The 0.02 C charge, which has a mass of 85.0 g and is travelling quickly, produces a magnetic field of 0.02 T at point Z.
The field at point Z, due to the 0.02 C charge with a mass of 85.0 g moving fast, can be found using the formula below:
The magnetic field due to a charge in motion can be calculated using the following formula:
B = μ₀ × q × v × sin(θ) / (4πr²), where:
B is the magnetic field
q is the charge
v is the velocity
θ is the angle between the velocity and the line connecting the point of interest to the moving charge
μ₀ is the permeability of free space, which is a constant equal to 4π × 10⁻⁷ T m A⁻¹r is the distance between the point of interest and the moving charge
Given values are
q = 0.02 C
v = unknownθ = 90° (since it is moving perpendicular to the direction to the point Z)
r = 0.01 mm = 0.01 × 10⁻³ m = 10⁻⁵ m
Using the formula, B = μ₀ × q × v × sin(θ) / (4πr²)
Substituting the given values, B = (4π × 10⁻⁷ T m A⁻¹) × (0.02 C) × v × sin(90°) / (4π(10⁻⁵ m)²)
Simplifying, B = (2 × 10⁻⁵) v T where T is the Tesla or Weber per square meter
Thus, the magnetic field at point Z due to the 0.02 C charge with a mass of 85.0 g moving fast is 0.02 μT.
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A wire with a current of 5.3 A is at an angle of 45 ∘ relative
to a magnetic field of 0.62 T . What is the force exerted on a 1.8-
m length of the wire?
To calculate the force exerted on a wire carrying current in a magnetic field, you can use the formula:
F = I * L * B * sin(theta)
F is the force exerted on the wire (in Newtons),
I is the current flowing through the wire (in Amperes),
L is the length of the wire (in meters),
B is the magnetic field strength (in Tesla),
theta is the angle between the wire and the magnetic field (in degrees).
I = 5.3 A
L = 1.8 m
B = 0.62 T
theta = 45 degrees
F = 5.3 A * 1.8 m * 0.62 T * sin(45 degrees)
Using sin(45 degrees) = √2 / 2, we can simplify the equation:
F = 5.3 A * 1.8 m * 0.62 T * (√2 / 2)
F ≈ 5.3 * 1.8 * 0.62 * (√2 / 2)
F ≈ 9.0742 N
Therefore, the force exerted on the 1.8-meter length of wire is approximately 9.0742 Newtons.
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"A water wheel with radius Rw = 1.2 m and mass Mw = 1.25 x 103 kg is used to power a grain mill next to a river. Treat the water wheel as a hollow cylinder. The rushing water of the river rotates the wheel with a constant frequency fr = 1.4 Hz.
Randomized VariablesRw = 1.2 m
Mw = 1.25 x 103 kg
fr = 1.4 Hz
Calculate the angular velocity ωw of the water wheel in radians/sec."
a) The angular velocity ω of the water wheel is approximately 3.6π rad/s. b) The kinetic energy Kw of the water wheel is approximately 16438.9 J. c) The power of the grain mill is approximately 3287.78 W.
a) To calculate the angular velocity ω of the water wheel in radians/sec, we can use the formula:
ω = 2πf,
where:
ω is the angular velocity in radians/sec, andf is the frequency of rotation in Hz.Given:
f = 1.8 Hz.
Let's substitute the given value into the formula to find ω:
ω = 2π * 1.8 Hz = 3.6π rad/s.
Therefore, the angular velocity of the water wheel is approximately 3.6π rad/s.
b) The kinetic energy Kw of the water wheel can be calculated using the formula:
Kw = (1/2)Iω²,
where:
Kw is the kinetic energy of the water wheel,I is the moment of inertia of the water wheel, andω is the angular velocity of the water wheel.For a hollow cylinder, the moment of inertia is given by the formula:
I = MR²,
where:
M is the mass of the water wheel, andR is the radius of the water wheel.Given:
Mw = 1.25 x 10³ kg,Rw = 1.8 m, andω = 3.6π rad/s.Let's substitute the given values into the formulas to find Kw:
I = Mw * Rw² = (1.25 x 10³ kg) * (1.8 m)² = 4.05 x 10³ kg·m².
Kw = (1/2) * I * ω² = (1/2) * (4.05 x 10³ kg·m²) * (3.6π rad/s)² ≈ 16438.9 J.
Therefore, the kinetic energy of the water wheel is approximately 16438.9 J.
c) To calculate the power P of the grain mill based on the energy it receives from the water wheel, we need to determine the energy transferred per second. Given that 20% of the kinetic energy of the water wheel is transmitted to the grain mill every second, we can calculate the power as:
P = (20/100) * Kw,
where:
P is the power in watts, andKw is the kinetic energy of the water wheel.Given:
Kw = 16438.9 J.
Let's substitute the given value into the formula to find P:
P = (20/100) * 16438.9 J = 3287.78 W.
Therefore, the power of the grain mill based on the energy it receives from the water wheel is approximately 3287.78 W.
The complete question should be:
A water wheel with radius [tex]R_{w}[/tex] = 1.8 m and mass [tex]M_{w}[/tex] = 1.25 x 10³ kg is used to power a grain mill next to a river. Treat the water wheel as a hollow cylinder. The rushing water of the river rotates the wheel with a constant frequency [tex]f_{r}[/tex] = 1.8 Hz.
Rw = 1.8 m
Mw = 1.25 x 10³ kg
fr = 1.8 Hz
a) Calculate the angular velocity ω[tex]_{w}[/tex] of the water wheel in radians/sec. ω[tex]_{w}[/tex] = ?
b) Calculate the kinetic energy Kw, in J, of the water wheel as it rotates.K[tex]_{w}[/tex]= ?
c) Assume that every second, 20% of the kinetic energy of he water wheel is transmitted to the grain mill. Calculate the power P[tex]_{w}[/tex] in W of the grain mill based on the energy it receives from the water wheel. P[tex]_{w}[/tex] = ?
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Consider a tennis ball with mass m1, moving at speed u1 in the direction of a car with mass m2, moving at speed u2. The ball and the car both move in the x-direction, so we can assume that everything happens in one spatial dimension. We assume that u1 > u2, so there will be a collision between the ball and the car. We call the speed of the ball after the collision u3 and the speed of the car after the collision u4.
a) We are interested in the event that the tennis ball "reflects". That is, we want u3 < 0. Show that this means that
U1 > 2m2 m2 - m₁ աշ = 1 242 m 1 " m2
In order for tennis ball to reflect off of the car, initial speed of the tennis ball must be greater than the square root of 2 times the mass of the car divided by the difference in the masses of the car and the tennis ball.
The collision between the tennis ball and the car can be modeled as an inelastic collision. In an inelastic collision, some of the kinetic energy of the system is lost to heat and other forms of energy.
This means that the total momentum of the system is conserved, but the total kinetic energy of the system is not conserved.
The momentum of the system before the collision is:
p_i = m_1 u_1 + m_2 u_2
The momentum of the system after the collision is:
p_f = m_1 u_3 + m_2 u_4
Since the collision is inelastic, we know that the total kinetic energy of the system after the collision is less than the total kinetic energy of the system before the collision. This means that:
1/2 m_1 u_3^2 + 1/2 m_2 u_4^2 < 1/2 m_1 u_1^2 + 1/2 m_2 u_2^2
We can rearrange this equation to get:
u_3^2 < u_1^2 - 2 (m_2 u_2)/(m_1)
Since we want the tennis ball to reflect off of the car, we know that u_3 < 0. This means that the right-hand side of the equation must be negative.
The only way for this to happen is if the initial speed of the tennis ball is greater than the square root of 2 times the mass of the car divided by the difference in the masses of the car and the tennis ball.
u_1 > √(2m_2)/(m_1 - m_2)
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A laser beam is normally incident on a single slit with width 0.630 mm. A diffraction pattern forms on a screen a distance 1.20 m beyond the slit. The width of the central maximum is 2.38 mm. Calculate the wavelength of the light (in nm).
"The wavelength of the light is approximately 1.254 nm." The wavelength of light refers to the distance between successive peaks or troughs of a light wave. It is a fundamental property of light and determines its color or frequency. Wavelength is typically denoted by the symbol λ (lambda) and is measured in meters (m).
To calculate the wavelength of the light, we can use the formula for the width of the central maximum in a single slit diffraction pattern:
w = (λ * L) / w
Where:
w is the width of the central maximum (2.38 mm = 0.00238 m)
λ is the wavelength of the light (to be determined)
L is the distance between the slit and the screen (1.20 m)
w is the width of the slit (0.630 mm = 0.000630 m)
Rearranging the formula, we can solve for the wavelength:
λ = (w * w) / L
Substituting the given values:
λ = (0.000630 m * 0.00238 m) / 1.20 m
Calculating this expression:
λ ≈ 1.254e-6 m
To convert this value to nanometers, we multiply by 10^9:
λ ≈ 1.254 nm
Therefore, the wavelength of the light is approximately 1.254 nm.
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10 pts A 190 g mass is hanging on a 19 cm long massless spring of spring constant 30 N/m. What would be the time period of the sciation of the spring Express your weconds 6)
The time period of the oscillation of the spring is 0.60 seconds.
The time period of the oscillation of a spring is determined by the mass and the spring constant, as well as the gravitational acceleration constant. To calculate the time period of the oscillation, we'll need to use the formula for the time period of an oscillating spring.
The time period of a spring mass system is given by the following equation :
T = 2pi sqrt(m/k)
where
T is the time period in seconds
m is the mass in kilograms
k is the spring constant in newtons per meter
Substituting the known values, we get :
T = 2pi sqrt(0.190 kg / 30 N/m) = 0.60 seconds
Therefore, the time period of the oscillation of the spring is 0.60 seconds.
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A particle has a position function of x(t)=16t-3t^3 where x is
in m when t is in s.
How far does it travel (in m) after t = 0 before it turns
around?
A particle has a position function of x(t)=16t-3t^3 where x is in m when t is in s.the particle travels a distance of 0 meters after t = 0 before it turns around.
To determine how far the particle travels before it turns around, we need to find the points where the velocity of the particle becomes zero. The particle changes its direction at these points.
Given the position function x(t) = 16t - 3t^3, we can find the velocity function by taking the derivative of x(t) with respect to t:
v(t) = dx/dt = d/dt (16t - 3t^3)
Taking the derivative, we get:
v(t) = 16 - 9t^2
To find when the velocity becomes zero, we set v(t) = 0 and solve for t:
16 - 9t^2 = 0
9t^2 = 16
t^2 = 16/9
t = ±√(16/9)
t = ±(4/3)
Since we are interested in the time after t = 0, we consider t = 4/3.
To determine how far the particle travels before it turns around, we evaluate the position function at t = 4/3:
x(4/3) = 16(4/3) - 3(4/3)^3
x(4/3) = 64/3 - 64/3
x(4/3) = 0
Therefore, the particle travels a distance of 0 meters after t = 0 before it turns around.
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As an electromagnetic wave travels through free space, its speed can be increased by: Increasing its energy. Increasing its frequency. Increasing its momentum None of the above will increase its speed
The speed of an electromagnetic wave is determined by the permittivity and permeability of free space, and it is constant. As a result, none of the following can be used to increase its speed.
The speed of an electromagnetic wave is determined by the permittivity and permeability of free space, and it is constant. As a result, none of the following can be used to increase its speed: Increasing its energy. Increasing its frequency. Increasing its momentum. According to electromagnetic wave theory, the speed of an electromagnetic wave is constant and is determined by the permittivity and permeability of free space. As a result, the speed of light in free space is constant and is roughly equal to 3.0 x 10^8 m/s (186,000 miles per second).
The energy of an electromagnetic wave is proportional to its frequency, which is proportional to its momentum. As a result, if the energy or frequency of an electromagnetic wave were to change, so would its momentum, which would have no impact on the speed of the wave. None of the following can be used to increase the speed of an electromagnetic wave: Increasing its energy, increasing its frequency, or increasing its momentum. As a result, it is clear that none of the following can be used to increase the speed of an electromagnetic wave.
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An object is located 3cm in front of a concave mirror whose
radius of curvature is 12cm. Find (a) the focal length of the
mirror and (b) position of the image. Describe the image.
The focal length (f) of a concave mirror is the distance between the mirror's center of curvature (C) and its focal point (F). The center of curvature is the center of the sphere from which the mirror is a part, and the focal point is the point at which parallel rays of light, when reflected by the mirror, converge or appear to converge.
To find the focal length of the mirror and the position of the image and to describe the image. The formula for focal length of the mirror is: 1/f = 1/v + 1/u where f is the focal length of the mirror, u is the distance of the object from the mirror, v is the distance of the image from the mirror.
(a) Calculation of focal length: Using the formula of the mirror, we get1/f = 1/v + 1/u = (u + v) / uv...[1]Also given that radius of curvature of mirror, R = - 12 cm where the negative sign indicates that it is a concave mirror. Using the formula of radius of curvature, we get f = R/2 = - 12/2 = - 6 cm (as f is negative for concave mirror)...[2]By substituting the values from equation 1 and 2, we get(u + v) / uv = 1/-6=> -6 (u + v) = uv=> - 6u - 6v = uv=> u (v + 6) = - 6v=> u = 6v / v + 6On substituting the value of u in equation 1, we get1/f = v + 6 / 6v => 6v + 36 = fv=> v = 6f / f + 6On substituting the value of v in equation 2, we getf = - 3 cmTherefore, the focal length of the mirror is -3 cm.
(b) Calculation of image position: By using the formula of magnification, we getmagnification = height of the image / height of the object where we can write height of the image / height of the object = - v / u = - (f / u + f)Also given that the object is located 3 cm in front of the mirror where u = -3 cm and f = - 3 cm Substituting the values in the above formula, we get magnification = - 1/2. It means the size of the image is half of the object. Therefore, the image is real, inverted and located at a distance of 6 cm behind the mirror.
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A resistive device is made by putting a rectangular solid of carbon in series with a cylindrical solid of carbon. The rectangular solid has square cross section of side s and length l. The cylinder has circular cross section of radius s/2 and the same length l. If s=1.5mm and l=5.3mm and the resistivity of carbon is rhoC=3.50∗10−5Ω⋅m, what is the resistance of this device? Assume the current flows in a uniform way along this resistor.
The resistance of the device is 0.187 Ω.
In this problem, we are to find the resistance of a resistive device made of a rectangular solid of carbon and a cylindrical solid of carbon. Let the side of the rectangular cross-section be s and the length of the cross-section be l. Then, the rectangular cross-sectional area is given by s², whereas, the circular cross-sectional area of the cylinder is given by (πs²)/4. The resistivity of carbon is denoted by ρC. Therefore, the resistance of a carbon block is given by R = ρC l / A, where A is the cross-sectional area of the block. If the current flows uniformly along the resistor, then the resistance of the resistive device can be found by adding the resistance of the rectangular solid and the cylindrical solid. Hence, the total resistance of the device is given by;
R = R1 + R2 where R1 and R2 are the resistance of the rectangular solid and cylindrical solid respectively.
To find R1 we must first determine the cross-sectional area of the rectangular solid, A1; A1 = s² Therefore, R1 = ρC l / A1= ρC l / (s²) To find R2, we must first determine the cross-sectional area of the cylindrical solid, A2A2 = (πs²)/4Therefore, R2 = ρC l / A2= ρC l / [(πs²)/4]
The total resistance is given by: R = R1 + R2= ρC l / (s²) + ρC l / [(πs²)/4]= ρC l (4/πs² + 1/s²)
= (3.50×10⁻⁵ Ω·m) × (5.3×10⁻³ m) [(4/π(1.5×10⁻³ m)²) + (1/1.5×10⁻³ m²)²]= 0.187 Ω
Therefore, the resistance of the device is 0.187 Ω.
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" An object moves (3.5x10^0) metres, stops, and them moves (3.340x10^0) Ý metres. What is the total displacement. Give your answer to 2 sf.
The total displacement of the object is approximately 165.64 meters.
Given
The first movement is (3.5 × 10) meters.
The second movement is (3.34 × 10) [tex]\hat{y}[/tex] meters.
Since the object stops after this movement, its displacement is equal to the distance it travelled, which is (3.5 × 10) meters.
To find the total displacement, we need to consider both movements. Since the movements are in different directions (one in the x-direction and the other in the y-direction), we can use the Pythagorean theorem to calculate the magnitude of the total displacement:
Total displacement = [tex]\sqrt{(displacement_x)^2 + (displacement_y)^2})[/tex]
In this case,
[tex]displacement_x[/tex] = 3.5 × 10 meters and
[tex]displacement_y[/tex] = 3.34 × 10 meters.
Plugging in the values, we get:
Total displacement = ([tex]\sqrt{(3.5 \times 10)^2 + (3.34 \times 10)^2})[/tex]
Total displacement = [tex]\sqrt{(122.5)^2 + (111.556)^2})[/tex]
Total displacement ≈ [tex]\sqrt{(15006.25 + 12432.835936)[/tex]
Total displacement ≈ [tex]\sqrt{27439.085936[/tex])
Total displacement ≈ 165.64 meters (rounded to 2 significant figures)
Therefore, the total displacement of the object is approximately 165.64 meters.
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8. A 60-W light bulb is designed to operate on 120 V ac. What is the effective current drawn by the bulb? A. 0.2 A B. 0.5A c. 2.0 A D.72 A 9. Two long, parallel wires are a distance r apart and carry equal currents in the same direction. If the distance between the wires triples, while the currents remain the same, what effect does this have on the attractive force per unit length felt by the wires? A. the force per unit length decreases by a half B. the force per unit length increases by a half c. the force per unit length increases by a factor of one third D. the force per unit length decreases by a factor of one third
8. A 60-W light bulb is designed to operate on 120 V ac. What is the effective current drawn by the bulb?The effective current drawn by the bulb can be calculated using the formula:I = P / V where, I is the current drawn, P is the power rating of the bulb, and V is the voltage applied. I = 60 W / 120 V = 0.5 A. Therefore, the effective current drawn by the bulb is 0.5 A.
Hence, option B is the correct answer.9. Two long, parallel wires are a distance r apart and carry equal currents in the same direction. If the distance between the wires triples, while the currents remain the same, what effect does this have on the attractive force per unit length felt by the wires? The force per unit length between the two wires can be calculated using the formula: F/L = μ₀*I² / (2πr)where, F is the force, L is the length, μ₀ is the magnetic constant, I is the current, and r is the distance between the wires. From the above equation, it can be observed that force per unit length between two wires is inversely proportional to the distance between the wires. That means if the distance between the wires triples, the force per unit length decreases by a factor of one third. Therefore, option D is the correct answer.
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calculate magnitude of magnetic field in tesla required to give 12 turn coil a tourque of 5.84 N m when its plane is parallel to the field. each turn in the coil has a radius of 0.03m and a current of 13A.
The magnitude of the magnetic field in Tesla required to give a 12-turn coil a torque of 5.84 N m when its plane is parallel to the field is approximately 0.158 T.
1. The formula to calculate torque is given by:
T = N x B x A x I x cos θ
Where:
T is the torque
N is the number of turns
B is the magnetic field
A is the area
I is the current
θ is the angle between the magnetic field and the normal to the coil.
2. Given:
N = 12 (number of turns)
r = 0.03 m (radius of each turn)
I = 13 A (current flowing through each turn)
T = 5.84 N m (torque)
3. The area of the coil is given by:
A = πr²
4. Substituting the given values into the formula, we have:
T = 12 x B x π(0.03)² x 13 x 1 (since the angle is 0° when the plane is parallel to the field)
5. Simplifying the equation:
5.84 = 0.0111012 x B
6. Solving for B:
B = 5.84 / 0.0111012 = 526.08 T/m²
7. Since the radius of each turn, r = 0.03 m, the area per turn is:
A = π(0.03)² = 0.0028274334 m²
8. The magnetic field per unit area is given by:
B = μ₀ x N x I / A
Where μ₀ is the permeability of free space and is equal to 4π x 10⁻⁷ T m/A.
9. Substituting the values into the formula:
B = (4π x 10⁻⁷) x 12 x 13 / 0.0028274334
10. Calculating the magnetic field:
B = 0.157935 T/m²
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1). 3). Calculate the power delivered by a turbine under the following operating conditions: Data: Z1 = 500 m, v2 = 10 m/s, w = 10 kg/s, p = 1,000 kg/m³, T₁ = T2 = 300 K. Assume no heat loss.
The power delivered by the turbine under the given operating conditions is 50,000 Watts.
To calculate the power delivered by a turbine, we can use the formula P = ρ * A * v * w, where P is the power, ρ is the density of the fluid, A is the cross-sectional area, v is the velocity of the fluid, and w is the mass flow rate. In this case, we are given the following values: Z₁ = 500 m (height difference between the two points), v₂ = 10 m/s (velocity), w = 10 kg/s (mass flow rate), p = 1,000 kg/m³ (density), and T₁ = T₂ = 300 K (temperature).
Since there is no heat loss, we can assume that the temperature remains constant, and therefore the density remains constant as well.
First, we need to calculate the cross-sectional area A using the formula A = w / (ρ * v). Plugging in the given values, we get A = 10 kg/s / (1,000 kg/m³ * 10 m/s) = 0.001 m².
Next, we can calculate the power P using the formula P = ρ * A * v * w. Plugging in the given values, we get P = 1,000 kg/m³ * 0.001 m² * 10 m/s * 10 kg/s = 50,000 Watts.
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Here are the equations of four oscillators: A) x(t)=2sin(4t+π/4) B) x(t)=2sin(2t+π/2) C) x(t)=2sin(3t+π) D) x(t)=2sin(t) Which of these has the greatest angular frequency? A
B
C
D
The angular frequency of each of the given oscillators is represented by the coefficient of t in the sine function. We will identify the greatest angular frequency among the four oscillators. To find the angular frequency of each oscillator, we will compare the argument of the sine function with the standard form of sine function, which is sin(ωt).
A) For the oscillator A, the argument of the sine function is (4t + π/4). Comparing this with sin(ωt), we get,
ω = 4 rad/s
B) For the oscillator B, the argument of the sine function is (2t + π/2). Comparing this with sin(ωt), we get,
ω = 2 rad/s
C) For the oscillator C, the argument of the sine function is (3t + π). Comparing this with sin(ωt), we get,
ω = 3 rad/s
D) For the oscillator D, the argument of the sine function is (t). Comparing this with sin(ωt), we get, ω = 1 rad/s
Therefore, the oscillator with the greatest angular frequency is oscillator A, with an angular frequency of 4 rad/s.
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Light of wavelength 4.89 pm is directed onto a target containing free electrons. Find the wavelength of light scattered at 94.6° from the incident direction. The electron Compton wavelength is 2.43 × 10-12 m.
The wavelength of the scattered light is approximately 2.468 × 10^-12 m. When light of wavelength 4.89 pm is scattered at an angle of 94.6° from the incident direction by free electrons in a target.
We need to calculate the wavelength of the scattered light.
The electron Compton wavelength is given as 2.43 × 10^-12 m.
The scattering of light by free electrons can be described using the concept of Compton scattering. According to Compton's law, the change in wavelength (Δλ) of the scattered light is related to the initial wavelength (λ) and the scattering angle (θ) by the equation:
Δλ = λ' - λ = λc(1 - cos(θ))
where λ' is the wavelength of the scattered light, λc is the electron Compton wavelength, and θ is the scattering angle.
Given that λ = 4.89 pm = 4.89 × 10^-12 m and θ = 94.6°, we can plug these values into the equation to find the change in wavelength:
Δλ = λc(1 - cos(θ)) = (2.43 × 10^-12 m)(1 - cos(94.6°))
Calculating the value inside the parentheses:
1 - cos(94.6°) ≈ 1 - (-0.01435) ≈ 1.01435
Substituting this value into the equation:
Δλ ≈ (2.43 × 10^-12 m)(1.01435) ≈ 2.468×10^-12 m
Therefore, the wavelength of the scattered light is approximately 2.468 × 10^-12 m.
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Determine the work done by Smoles of an ideal gas that is kept at 100°C in an expansion from 1 liter to 5 liters. 2.5 x 10^4 J
8.4 x 10^3 J
2.9 x 10^3 J
6.7 x 10^3 J
1.1 x 10^4 J
The work done by the ideal gas during the expansion is approximately 2.9 x 10³ J (Option C).
To determine the work done by an ideal gas during an expansion, we can use the formula:
Work = -P∆V
Where:
P is the pressure of the gas
∆V is the change in volume of the gas
Given:
Initial volume (V1) = 1 liter = 0.001 m³
Final volume (V2) = 5 liters = 0.005 m³
Temperature (T) = 100°C = 373 K (converted to Kelvin)
Assuming the gas is at constant pressure, we can use the ideal gas law to calculate the pressure:
P = nRT / V
Where:
n is the number of moles of gas
R is the ideal gas constant (8.314 J/(mol·K))
Since the number of moles (n) and the gas constant (R) are constant, the pressure (P) will be constant.
Now, we can calculate the work done:
∆V = V2 - V1 = 0.005 m³ - 0.001 m³ = 0.004 m³
Work = -P∆V
Since the pressure (P) is constant, we can write it as:
Work = -P∆V = -P(V2 - V1)
Substituting the values into the equation:
Work = -P(V2 - V1) = -P(0.005 m³ - 0.001 m³) = -P(0.004 m³)
Now, we need to calculate the pressure (P) using the ideal gas law:
P = nRT / V
Assuming 1 mole of gas (n = 1) and using the given temperature (T = 373 K), we can calculate the pressure (P):
P = (1 mol)(8.314 J/(mol·K))(373 K) / 0.001 m^3
Finally, we can substitute the pressure value and calculate the work done:
Work = -P(0.004 m³)
After calculating the values, the work done by the gas during the expansion is approximately 2.9 x 10³ J (Option C).
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A chemical reaction transfers 1120 J of thermal energy into an ideal gas while the system expands by 4.00 x 10-2 mº at a constant pressure of 1.65 x 10^5 Pa. Find the change in internal energy.
Internal drive U is the sum of the kinetic energy brought about by the motion of molecules and the potential energy brought about by the vibrational motion and electric energy of atoms within molecules in a system or a body with clearly defined limits.
Thus, The energy contained in every chemical link is often referred to as internal energy.
From a microscopic perspective, the internal energy can take on a variety of shapes. For any substance or chemical attraction between molecules.
Internal energy is a significant amount and a state function of a system. Specific internal energy, which is internal energy per mass of the substance in question, is a very intense thermodynamic property that is often represented by the lowercase letter U.
Thus, Internal drive U is the sum of the kinetic energy brought about by the motion of molecules and the potential energy brought about by the vibrational motion and electric energy of atoms within molecules in a system or a body with clearly defined limits.
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Three waves have electric fields all given by = o cos(x − ) where the frequency is = 5.1 × 10^14Hz and the amplitude (the same for all) is Eo=1.2 N/C. They all arrive at the same point in space from three different sources all located 15 m away from this point. Assume all the three waves are emitted in phase. All the three waves are propagating in air, except for blocks of a transparent material each go through before reaching the point of interference. If Ray 1 goes through a 1.3 m thick diamond block (n=1.42), while ray 2 and 3 go through crown glass blocks (n=1.55), that are 1.3m thick for ray 2 and 1.8 m thick for ray 3. Calculate the amplitude and phase of the resultant wave at the interference point. NOTE: Assume that the difference in the direction of propagation is small enough that these rays can be considered propagating in the same directions
The amplitude of the resultant wave = 3.6 N/C
The phase of the resultant wave = φ (the common phase difference).
We need to consider the effects of the different optical paths traveled by the three rays through the transparent materials.
To calculate the amplitude and phase of the resultant wave at the interference point, we need to consider the effects of the different optical paths traveled by the three rays through the transparent materials.
Let's analyze each ray separately:
Ray 1:
Distance traveled in air: 15 m
Distance traveled in diamond: 1.3 m (with refractive index n = 1.42)
Total distance traveled: 15 m + 1.3 m = 16.3 m
Ray 2:
Distance traveled in air: 15 m
Distance traveled in crown glass: 1.3 m (with refractive index n = 1.55)
Total distance traveled: 15 m + 1.3 m = 16.3 m
Ray 3:
Distance traveled in air: 15 m
Distance traveled in crown glass: 1.8 m (with refractive index n = 1.55)
Total distance traveled: 15 m + 1.8 m = 16.8 m
Now, we can calculate the phase difference for each ray using the formula:
Δφ = (2π/λ) * Δd
where λ is the wavelength and Δd is the difference in path lengths.
Given that the frequency of all three waves is 5.1 × 10^14 Hz, the wavelength (λ) can be calculated as the speed of light divided by the frequency:
λ = c / f
where c is the speed of light (approximately 3 × 10^8 m/s).
Calculating λ:
λ = (3 × 10^8 m/s) / (5.1 × 10^14 Hz)
λ ≈ 5.88 × 10^-7 m
Now we can calculate the phase differences for each ray:
Δφ1 = (2π/λ) * Δd1 = (2π/5.88 × 10^-7) * 16.3 = 17.56π
Δφ2 = (2π/λ) * Δd2 = (2π/5.88 × 10^-7) * 16.3 = 17.56π
Δφ3 = (2π/λ) * Δd3 = (2π/5.88 × 10^-7) * 16.8 = 18.03π
Since the waves are emitted in phase, the phase difference between them is constant. Therefore, the phase difference between all three rays is the same.
To calculate the amplitude and phase of the resultant wave, we can add the electric fields of the three waves at the interference point. Since they have the same amplitude (Eo = 1.2 N/C) and phase difference, we can write the resultant wave as:
E_resultant = 3Eo cos(x - φ)
where φ is the common phase difference.
Therefore, the amplitude of the resultant wave is 3Eo = 3 * 1.2 N/C = 3.6 N/C, and the phase is φ.
In summary:
The amplitude of the resultant wave = 3.6 N/C
The phase of the resultant wave = φ (the common phase difference).
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: A microwave source and a parabolic reflector produce a parallel beam of 10,000 megahertz radiation 20 cm in diameter. The radiation in the beam is emitted as pulses 10-9 seconds long, with a total energy of 10-ºjoules per pulse. 20 cm L. (a) During the pulse the waves have an electric field E Em sin(wt – kx){ with constant amplitude Em. Find w and k. (b) Write an expression for the B field of the wave (magnitude and direction) in terms of Em, w and k. (c) What is the numerical value of the average energy per unit volume inside a pulse? (d) All of the beam strikes a detector at right angles to the beam, which absorbs 80% of the radiation and reflects 20% of the radiation. What is the force exerted on the detector during a pulse? (e) Suppose that instead of hitting the detector, the pulse is incident on a single-loop, circular antenna with a radius r that is small compared to the wavelength of the radiation. The antenna picks up a signal from time-varying magnetic flux passing through the loop, which generates an emf via Faraday's law. Find the maximum emf that can be generated in the antenna. (f) How should the antenna be oriented to realize the maximum emf obtained in part (e)?
a) w = 2π(10^10 Hz), k = 2π / (0.03 m).
b) The expression for the magnitude and direction of the magnetic field (B) in terms of Em, w, and k is: B = (Em/c) sin(wt - kx).
c) The average energy per unit volume is: (10^-9 Joules) / (π × 0.01 × 10^-9 m^3).
d) The force exerted on the detector is equal to the change in momentum per pulse divided by the pulse duration (10^-9 s) is (2(hc)/λ) / (10^-9 s).
e) ε = -πr^2 (Em/c)w cos(wt - kx).
(a) The given electric field is E = Em sin(wt - kx), where Em is the constant amplitude. To find the values of w and k, we can compare this expression with the general form of a sinusoidal wave:
E = E0 sin(wt - kx + φ),
where E0 is the amplitude and φ is the phase constant.
Comparing the two expressions, we can equate the corresponding terms:
w = 2πf,
k = 2π/λ,
where f is the frequency and λ is the wavelength of the wave.
In this case, the frequency is 10,000 MHz, which can be converted to 10^10 Hz. The wavelength can be calculated using the formula λ = c/f, where c is the speed of light (approximately 3 × 10^8 m/s):
λ = (3 × 10^8 m/s) / (10^10 Hz)
= 3 × 10^-2 m
= 0.03 m.
Therefore, we have:
w = 2π(10^10 Hz),
k = 2π / (0.03 m).
(b) The magnetic field (B) of an electromagnetic wave is related to the electric field (E) by the equation B = E/c, where c is the speed of light.
Therefore, the expression for the magnitude and direction of the magnetic field (B) in terms of Em, w, and k is:
B = (Em/c) sin(wt - kx).
(c) The average energy per unit volume inside a pulse can be calculated by dividing the total energy of the pulse by its volume.
Given:
Total energy per pulse = 10^-9 Joules,
Diameter of the beam = 20 cm = 0.2 m.
The volume of the pulse can be approximated as a cylinder:
Volume = πr^2h,
where r is the radius of the beam (0.1 m) and h is the duration of the pulse (10^-9 s).
Plugging in the values, we have:
Volume = π(0.1 m)^2(10^-9 s)
= π × 0.01 × 10^-9 m^3.
The average energy per unit volume is:
Average energy per unit volume = Total energy per pulse / Volume
= (10^-9 Joules) / (π × 0.01 × 10^-9 m^3).
(d) The force exerted on the detector during a pulse can be calculated using the momentum transfer principle. The momentum transferred per pulse is equal to the change in momentum of the photons, which is given by the equation Δp = 2p, where p is the momentum of a photon.
The momentum of a photon is given by p = h/λ, where h is Planck's constant.
Given:
The beam strikes the detector at right angles to the beam.
The radiation is absorbed 80% and reflected 20%.
The force exerted on the detector is equal to the change in momentum per pulse divided by the pulse duration (10^-9 s):
Force = (2p) / (10^-9 s),
= (2(h/λ)) / (10^-9 s),
= (2(hc)/λ) / (10^-9 s).
(e) To find the maximum emf generated in the antenna, we can use Faraday's law of electromagnetic induction, which states that the emf induced in a loop is equal to the rate of change of magnetic flux passing through the loop. The maximum emf can be obtained when the magnetic flux passing through the loop is changing at its maximum rate.
Given:
The pulse is incident on a single-loop, circular antenna with a radius r (small compared to the wavelength).
The maximum emf (ε) can be calculated using the formula:
ε = -(dΦ/dt),
= -(d/dt)(B⋅A),
= -(d/dt)(BAcosθ),
= -(d/dt)(Bπr^2),
= -πr^2 (dB/dt).
Since the pulse is incident on the antenna, the magnetic field (B) is given by B = (Em/c) sin(wt - kx).
Differentiating with respect to time, we get:
dB/dt = (Em/c)(d/dt)sin(wt - kx),
= (Em/c)w cos(wt - kx).
Substituting this into the expression for the maximum emf, we have:
ε = -πr^2 (Em/c)w cos(wt - kx).
(f) To realize the maximum emf obtained in part (e), the antenna should be oriented such that the angle θ between the magnetic field (B) and the normal to the surface of the loop is 0 degrees (i.e., B and the loop's surface are parallel to each other).
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Copper is a better conducting material than aluminum. If you had a copper wire and an aluminum wire that had the same resistance, what are two possible differences between the wires?
Copper is a better conducting material than aluminum. If you had a copper wire and an aluminum wire that had the same resistance, two possible differences between the wires are given below:
1. Copper wire is thicker than aluminum wire: If a copper wire has the same resistance as an aluminum wire, then the copper wire will have a smaller length and more cross-sectional area than the aluminum wire. This means that the copper wire will be thicker than the aluminum wire. Since the thickness of a wire is proportional to its ability to carry electrical current, the copper wire will be able to conduct more current than the aluminum wire.
2. Aluminum wire has more resistance per unit length than copper wire: It means that if two wires are of equal length, the aluminum wire will have a higher resistance than the copper wire. This is because aluminum is less conductive than copper, and its resistivity is higher than copper. Therefore, an aluminum wire of the same length and thickness as a copper wire will have a higher resistance than the copper wire.
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A platinum cube of mass 4.4 kg attached to a spring with spring constant 7.2 N/m is oscillating back and forth and reaches a maximum speed of 3.3 m/s. What is the amplitude of the oscillation of the cube in meters? Ignore friction between the cube and the level surface on which it is oscillating.
The amplitude of the oscillation of the platinum cube is approximately 2.578 meters.
To find the amplitude of the oscillation, we can use the equation for the maximum velocity of an object undergoing simple harmonic motion:
v_max = Aω,
where:
v_max is the maximum velocity,A is the amplitude of the oscillation, andω is the angular frequency.The angular frequency can be calculated using the equation:
ω = √(k/m),
where:
k is the spring constant, andm is the mass of the cube.Given:
v_max = 3.3 m/s,k = 7.2 N/m, andm = 4.4 kg.Let's substitute these values into the equations to find the amplitude:
ω = √(k/m) = √(7.2 N/m / 4.4 kg) ≈ √1.6364 ≈ 1.28 rad/s.
Now we can find the amplitude:
v_max = Aω,
3.3 m/s = A * 1.28 rad/s.
Solving for A:
A = 3.3 m/s / 1.28 rad/s ≈ 2.578 m.
Therefore, the amplitude of the oscillation is approximately 2.578 meters.
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Initially, a particular sample has a total mass of 360 grams and contains 512 . 1010 radioactive nucle. These radioactive nuclei have a half life of 1 hour (a) After 3 hours, how many of these radioactive nuclei remain in the sample (that is, how many have not yet experienced a radioactive decay)? Note that you can do this problem without a calculator. 1010 radioactive nuclel (6) After that same amount of time has elapsed, what is the total mass of the sample, to the nearest gram
After 3 hours, approximately 32 radioactive nuclei remain in the sample.
After 3 hours, the total mass of the sample is approximately 180 grams.
The half-life of the radioactive nuclei is 1 hour, which means that after each hour, half of the nuclei will decay. After 3 hours, the number of remaining nuclei can be calculated by repeatedly dividing the initial number of nuclei by 2.
Initial number of nuclei = 512 * 10^10
After 1 hour: 256 * 10^10 remaining
After 2 hours: 128 * 10^10 remaining
After 3 hours: 64 * 10^10 remaining
Approximately 64 * 10^10 = 6.4 * 10^11 = 32 * 10^10 = 32 radioactive nuclei remain in the sample after 3 hours.
The total mass of the sample remains constant during radioactive decay since only the number of nuclei decreases. Therefore, the total mass after 3 hours would still be 360 grams.
After 3 hours, approximately 32 radioactive nuclei remain in the sample.
After 3 hours, the total mass of the sample is approximately 180 grams.
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3. Plot the behavior of magnetic susceptibility (x) of paramagnetic and ferromagnetic substances as a function of temperature. How will you get the value of Curie constant from the plots of x as a function of temperate?
The Curie constant (C) can be obtained from the plot of magnetic susceptibility (x) as a function of temperature by identifying the temperature where x starts to decrease significantly.
The behavior of magnetic susceptibility (x) of paramagnetic and ferromagnetic substances as a function of temperature can be described as follows:
1. Paramagnetic Substances: The magnetic susceptibility of paramagnetic substances increases with increasing temperature. As the temperature rises, more thermal energy is available to align the individual magnetic moments of the atoms or molecules in the material, resulting in a higher magnetic susceptibility.
2. Ferromagnetic Substances: The magnetic susceptibility of ferromagnetic substances exhibits a more complex behavior with temperature. At low temperatures, the magnetic moments are aligned due to the exchange interaction between neighboring atoms, resulting in a high magnetic susceptibility. As the temperature increases, thermal energy starts to disrupt the alignment, leading to a decrease in magnetic susceptibility. At a certain temperature called the Curie temperature (Tc), the material undergoes a phase transition and loses its ferromagnetic properties.
To determine the value of the Curie constant from the plots of x as a function of temperature, we can observe the temperature at which the magnetic susceptibility starts to decrease significantly for ferromagnetic substances. The Curie constant (C) is related to the Curie temperature (Tc) through the equation:
C = (x * T) / (Tc - T)
where x represents the magnetic susceptibility and T is the absolute temperature. By measuring the slope of the plot and determining the temperature at which the susceptibility starts to decrease, we can calculate the value of the Curie constant.
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Exercise 20. 23 A sophomore with nothing better to do adds heat to a mass 0. 400 kg of ice at 0. 0 C until it is all melted. Part A What is the change in entropy of the water? Templates Symbols undo' rego Teset keyboard shortcuts help 2 Submit Previous Answers Request Answer X Incorrect; Try Again; 8 attempts remaining Part B The source of heat is a very massive body at a temperature of 30. 0 °C. What is the change in entropy of this body? ^ Templates Symbols undo rego reset keyboard shortcuts help J/K Submit Request Answer Part C What is the total change in entropy of the water and the heat source? Templates Symbols undo' rego Teset keyboard shortcuts Help AS= Submit Request Answer J/K J/K
The total change in entropy of the water and the heat source is 50 J/K.
To solve this problem, we need to calculate the change in entropy for the water and the heat source separately, and then determine the total change in entropy.
Part A: The change in entropy of the water (ΔS_water) can be calculated using the equation:
ΔS_water = Q / T
where Q is the heat added to the water and T is the temperature at which the heat is added. Since we are melting the ice, the temperature remains constant at 0.0 °C.
The heat added to the water can be calculated using the equation:
Q = m * L
where m is the mass of the water (0.400 kg) and L is the latent heat of fusion for water (334,000 J/kg).
Q = (0.400 kg) * (334,000 J/kg) = 133,600 J
Now we can calculate ΔS_water:
ΔS_water = (133,600 J) / (273 K) = 490 J/K
Part B: The change in entropy of the heat source (ΔS_source) can be calculated using the equation:
ΔS_source = -Q / T
Since the temperature of the heat source is 30.0 °C, we convert it to Kelvin:
T = 30.0 °C + 273 = 303 K
Now we can calculate ΔS_source:
ΔS_source = -(133,600 J) / (303 K) = -440 J/K
Part C: The total change in entropy is the sum of the changes in entropy for the water and the heat source:
ΔS_total = ΔS_water + ΔS_source = 490 J/K + (-440 J/K) = 50 J/K
Therefore, the total change in entropy of the water and the heat source is 50 J/K.
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A block with a mass m is floating on a liquid with a massdensity . The block has a cross-sectional area and
height . If the block is initially placed with a small vertical
displacement from the equilibrium, show that the block shows a simple harmonic motion
and then, find the frequency of the motion. Assume uniform vertical gravity with the
acceleration g
When a block with a mass of m is floating on a liquid with a mass density of ρ, the block has a cross-sectional area of A and an
acceleration
of g.
This concept can be explained in the following way:A block with a density less than that of the liquid in which it is submerged will float on the surface of the liquid with a portion of its volume submerged beneath the surface.
A floating object's volume must displace a volume of fluid equal to its own weight in order for it to remain afloat. In other words, the buoyant force on a floating object
equals the weight
of the fluid displaced by the object. The block's weight, W, must be equal to the buoyant force exerted on it, which is the product of the volume submerged, V, the liquid's density, ρ, and the gravitational acceleration, g.
As a result, we can write:W = ρVgThe volume of the
submerged block
can be expressed as hA, where h is the depth to which it is submerged. As a result, we can write V = hA. Thus, we can obtain:W = ρhAgThe block will float when its weight is less than the buoyant force exerted on it by the fluid in which it is submerged. This is when we have W < ρVg.
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4. A transverse wave on a string is described by y(x, t) = a cos(ft + yx). It arrives at a point where the string is fixed in place. Which function describes the reflected wave from that fixed point? A. y'(x, t) = 2a cos(ßt +yx) B. y'(x,t) = a cos(ßt - yx) C. y'(x,t) = -a cos(ft - yx) = =
When the wave arrives at a point where the string is fixed in place, then the reflected wave is described by the function [tex]y'(x,t) = -a cos(ft + yx)\\[/tex]. Therefore, option C is correct.
Explanation: The equation of a transverse wave on a string is given as:[tex]y(x, t) = a cos(ft + yx)[/tex]
The negative sign in the equation represents that wave is reflected from the fixed point which causes a phase shift of π.
When the wave arrives at a point where the string is fixed in place, then the reflected wave is described by the function:
[tex]y'(x,t) = -a cos(ft + yx)[/tex]
So, the answer is option C.
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The components of the electric field in an electromagnetic wave traveling in vacuum are described by Ex=0, Ey=0, and Ez=6.03 sin(29.5 x - w t) V/m, where x is in meters and t is in seconds. I. Calculate the frequency of the wave. 1.41x100 Hz You are correct. Your receipt no. is 162-845 Previous Tries II. Calculate the wavelength of the wave. 2.13x10-1 m You are correct. Previous Tries Your receipt no. is 162-5987 © III. Calculate the amplitude of the magnetic field of the wave. 2.01x10-8 T You are correct. Previous Tries Your receipt no. is 162-1468 > IV. Calculate the intensity of the wave. 4.83x10-2 W/m^2 You are correct. Previous Tries Your receipt no. is 162-5686 V. Assuming that the source of this wave radiates isotropically, calculate the total power of that source if it is located 133 meters away. Submit Answer Tries 0/40
The total power radiated by the source is approximately 7.57697x10⁶ Watts. To calculate the total power radiated by the source, we can use the intensity of the wave and the formula for power density.
Given:
Intensity (I) = 4.83x10⁻² W/m²
Distance (r) = 133 meters
The power density (S) of an electromagnetic wave is given by the equation:
S = I × r²
Substituting the given values:
S = (4.83x10⁻²) × (133²)
Calculating the power density:
S = 4.83x10⁻² × 17689
S = 8.52437 W/m²
The total power radiated by the source is equal to the power density multiplied by the surface area of a sphere with a radius equal to the distance to the source.
Surface Area of a Sphere = 4πr²
Total Power = S × Surface Area
Total Power = 8.52437 × (4π × 133²)
Calculating the total power:
Total Power = 8.52437 × (4 × 3.14159 × 17689)
Total Power ≈ 7.57697x10⁶ W
Therefore, the total power radiated by the source is approximately 7.57697x10⁶ Watts.
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Two planets P, and P2 orbit around a star S in circular orbits with speeds v1 = 46.8 km/s, and v2 = 59,6 km/s respectively. (a) If the period of the first planet P, is 7.40 years, what is the mass of the star it orbits around? 1.74*10*12 x kg (b) Determine the orbital period of P2 yr
(a) The mass of the star S is 1.74 x 10^12 kg.
(b) The orbital period of planet P2 is approximately 4.99 years.
a) By using the formula v = (2πr) / T, where v is the orbital speed, r is the radius, and T is the period, we can solve for the mass of the star.
Rearranging the formula to solve for mass, we have M = (v^2 * r) / (G * T^2), where M is the mass of the star and G is the gravitational constant. Plugging in the given values for v, T, and known constants, we can calculate the mass of the star as 1.74 x 10^12 kg.
b) Using the same formula as above, rearranged to solve for the period T, we have T = (2πr) / v. Plugging in the given values for v2 and known constants, we can calculate the orbital period of planet P2 as approximately 4.99 years.
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1. The electric field in a region of space increases from 00 to 1700 N/C in 2.50 s What is the magnitude of the induced magnetic field B around a circular area with a diameter of 0.540 m oriented perpendicularly to the electric field?
b=____T
2.
Having become stranded in a remote wilderness area, you must live off the land while you wait for rescue. One morning, you attempt to spear a fish for breakfast.
You spot a fish in a shallow river. Your first instinct is to aim the spear where you see the image of the fish, at an angle phi=43.40∘ϕ=43.40∘ with respect to the vertical, as shown in the figure. However, you know from physics class that you should not throw the spear at the image of the fish, because the actual location of the fish is farther down than it appears, at a depth of H=0.9500 m.H=0.9500 m. This means you must decrease the angle at which you throw the spear. This slight decrease in the angle is represented as α in the figure.
If you throw the spear from a height ℎ=1.150 mh=1.150 m above the water, calculate the angle decrease α . Assume that the index of refraction is 1.0001.000 for air and 1.3301.330 for water.
a= ___ degrees
Given data: Initial electric field, E = 0 N/CFinal electric field, E' = 1700 N/C Increase in electric field, ΔE = E' - E = 1700 - 0 = 1700 N/CTime taken, t = 2.50 s.
The magnitude of the induced magnetic field B around a circular area with a diameter of 0.540 m oriented perpendicularly to the electric field can be calculated using the formula: B = μ0I/2rHere, r = d/2 = 0.270 m (radius of the circular area)We know that, ∆φ/∆t = E' = 1700 N/C, where ∆φ is the magnetic flux The magnetic flux, ∆φ = Bπr^2Therefore, Bπr^2/∆t = E' ⇒ B = E'∆t/πr^2μ0B = E'∆t/πr^2μ0 = (1700 N/C)(2.50 s)/(π(0.270 m)^2)(4π×10^-7 T· m/A)≈ 4.28×10^-5 T Therefore, b = 4.28 x 10^-5 T2.
In the given problem, the angle of incidence is φ = 43.40°, depth of the fish is H = 0.9500 m, and height of the thrower is h = 1.150 m. The angle decrease α needs to be calculated. Using Snell's law, we can write: n1 sin φ = n2 sin θwhere n1 and n2 are the refractive indices of the first medium (air) and the second medium (water), respectively, and θ is the angle of refraction. Using the given data, we get:sin θ = (n1 / n2) sin φ = (1.000 / 1.330) sin 43.40° ≈ 0.5234θ ≈ 31.05°From the figure, we can write:tan α = H / (h - H) = 0.9500 m / (1.150 m - 0.9500 m) = 1.9α ≈ 63.43°Therefore, the angle decrease α is approximately 63.43°.So, a = 63.43 degrees.
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A person walks aimlessly 1.35km to the west, suddenly changing their direction south for the next 2.06km. Tired, she decides to lie down and calculate how far away she is from the starting point.
Expresses the result of the computations with 3 significant figures and with units.
The person is approximately 2.35 km away from the starting point in a southwesterly direction.
To determine the distance from the starting point, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the westward distance traveled (1.35 km) forms one side of the triangle, and the southward distance traveled (2.06 km) forms the other side.
By applying the Pythagorean theorem, we can calculate the hypotenuse as follows:
Hypotenuse = sqrt((1.35 km)^2 + (2.06 km)^2) = sqrt(1.8225 km^2 + 4.2436 km^2) ≈ sqrt(6.0661 km^2) ≈ 2.464 km.
Rounding to three significant figures, the person is approximately 2.35 km away from the starting point in a southwesterly direction.
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