A rectangular wire loop with current i=200mA, width W=20cm and length L=50cm is placed in a uniform magnetic field B=7T as shown. The normal to the loop, represented by the dashed line, is θ=20degrees from the magnetic field direction.
#3A (6 points possible)
What is the magnitude of the torque on the loop in N−m?
What is the magnitude of the torque on the loop in N−m? - unanswered
What is the direction?
- unanswered right (along B) up left down into page out of page
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#3B (4 points possible)
What is the largest difference in potential energy in Joules between any two orientations of the loop in the uniform field (meaning any values of θ)? Enter a positive quantity.
What is the largest difference in potential energy in Joules between any two orientations of the loop in the uniform field (meaning any values of θ)? Enter a positive quantity. - unanswered
Check your answer Save your answer You have used 0 of 3 submissions
The direction of torque is down.
the largest difference in potential energy is 0.28 J, it happens when θ = 180° (maximum) and θ=0° (minimum).
To find the magnitude of the torque on the loop, we can use the formula:
τ = NIABsinθ
where τ is the torque, N is the number of turns in the loop, I is the current, A is the area of the loop, B is the magnetic field, and θ is the angle between the magnetic field and the normal to the loop.
Given:
I = 200 mA = 200 x 10^-3 A
W = 20 cm = 20 x 10^-2 m
L = 50 cm = 50 x 10^-2 m
B = 7 T
θ = 20 degrees
First, let's calculate the area of the loop:
A = L x W
= (50 x 10^-2 m) x (20 x 10^-2 m)
= 1000 x 10^-4 m^2
= 10^-1 m^2
Next, let's calculate the torque:
τ = NIABsinθ
= (1)(200 x 10^-3 A)(10^-1 m^2)(7 T)(sin(20 degrees))
≈ 2 x 10^-1 N-m (rounded to one decimal place)
Therefore, the magnitude of the torque on the loop is approximately 0.2 N-m.
Now, let's move on to finding the largest difference in potential energy between any two orientations of the loop.
The potential energy of a magnetic dipole in a uniform magnetic field is given by the formula:
U = -μBcosθ
where U is the potential energy, μ is the magnetic dipole moment, B is the magnetic field, and θ is the angle between the magnetic field and the dipole moment.
In this case, the magnetic dipole moment is given by:
μ = NIA
where N is the number of turns, I is the current, and A is the area of the loop.
Since we are looking for the largest difference in potential energy between any two orientations, we can assume that the magnetic dipole moment remains constant.
Therefore, to find the largest difference in potential energy, we need to find the maximum and minimum values of cosθ.
Since cosθ is the maximum when θ = 0 degrees (parallel to the magnetic field) and the minimum when θ = 180 degrees (anti-parallel to the magnetic field), we can calculate the potential energy at both positions:
U_max = -μBcos(0 degrees)
U_min = -μBcos(180 degrees)
Substituting the values:
U_max = -(NIA)Bcos(0 degrees)
U_min = -(NIA)Bcos(180 degrees)
Now, let's substitute the expression for μ:
U_max = -(NIA)Bcos(0 degrees)
= -(NIA)B(1)
= -NIA(B)
U_min = -(NIA)Bcos(180 degrees)
= -(NIA)B(-1)
= NIA(B)
Taking the absolute values of U_max and U_min:
|U_max| = NIA(B)
|U_min| = NIA(B)
The largest difference in potential energy between any two orientations is given by:
ΔU = |U_max| - |U_min|
= NIA(B) - NIA(B)
= 2NIA(B)
Let's substitute the given values:
ΔU = 2(1)(200 x 10^-3 A)(10^-1 m^2)(7 T)
= 2 x 10^-1 J
Therefore, the largest difference in potential energy between any two orientations of the loop in the uniform field is approximately 0.2 J.
To find the magnitude of the torque on the loop, we can use the equation:
τ = NIABsinθ
Where τ is the torque, N is the number of turns in the loop, I is the current, A is the area of the loop, B is the magnetic field strength, and θ is the angle between the normal to the loop and the magnetic field direction.
In this case, we are given that the current is 200mA, the width of the loop is 20cm, the length of the loop is 50cm, the magnetic field is 7T, and the angle θ is 20 degrees.
First, we need to calculate the area of the loop. Since it is rectangular, the area is simply the product of the width and the length: A = (20cm) * (50cm) = 1000cm^2 = 0.1m^2.
Next, we plug in the values into the torque equation and solve for the torque:
τ = (1)(0.2A)(7T)sin(20°)
= 0.2 * 0.1m^2 * 7T * sin(20°)
≈ 0.2 * 0.1 * 7 * 0.342
≈ 0.048 N-m
Therefore, the magnitude of the torque on the loop is approximately 0.048 N-m.
As for the direction of the torque, it can be determined using the right-hand rule. If you align your right-hand thumb with the magnetic field direction (B), and curl your fingers towards the loop, the direction your fingers point will indicate the direction of the torque. In this case, it should be out of the page or towards you.
Now let's move on to the next question - the largest difference in potential energy between any two orientations of the loop in the uniform field.
The potential energy of the loop in a magnetic field can be calculated using the equation:
U = -mBcosθ
Where U is the potential energy, m is the magnetic moment of the loop, B is the magnetic field strength, and θ is the angle between the magnetic moment and the magnetic field direction.
To find the largest difference in potential energy between any two orientations of the loop, we need to find the maximum and minimum potential energy values by considering the extreme angles.
In this case, since the magnetic moment of the loop is equal to the current times the area of the loop (m = IA), we can use the same values that we calculated earlier: I = 0.2A and A = 0.1m^2.
First, let's calculate the potential energy at the maximum angle, which is 90 degrees:
U_max = -(0.2A)(7T)cos(90°)
= -0.2 * 0.1m^2 * 7T * cos(90°)
= 0J
Next, let's calculate the potential energy at the minimum angle, which is 0 degrees:
U_min = -(0.2A)(7T)cos(0°)
= -0.2 * 0.1m^2 * 7T * cos(0°)
= -0.14J
Therefore, the largest difference in potential energy between any two orientations of the loop in the uniform field is 0J - (-0.14J) = 0.14J.
In summary, the magnitude of the torque on the loop is approximately 0.048 N-m, and the largest difference in potential energy between any two orientations of the loop in the uniform field is 0.14J.