A current loop with radius 20 cm and current 2 A is in a uniform magnetic field of 0.5 T. Considering all possible orientations of the loop relative to the field, what is the largest potential energy difference (in Joules) you can find between two orientations?

p=IA= Iπr²

τ=pBsinφ (sinφ=1) =>
τ=pB

A current of 20 mA flows in a single circular loop with a radius of 2 meters. A uniform magnetic field of 1.2 T points parallel to the plane of the loop. What is the magnitude of the magnetic moment of the loop in A-m^2?

What is the magnitude of the torque on the loop in N-m?

To find the largest potential energy difference between two orientations of the loop, we need to calculate the potential energy for both orientations and then calculate the difference.

The potential energy (U) of a current loop in a magnetic field is given by the equation:

U = -m * B * cos(θ)

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.

The magnetic moment (m) of a loop is defined as the product of the current (I) flowing through the loop and the area (A) enclosed by the loop. In this case, the area of the loop can be calculated as:

A = π * r^2

where r is the radius of the loop.

Let's calculate the potential energy for both orientations of the loop.

1. Orientation 1: When the magnetic moment of the loop is perpendicular to the magnetic field.

In this orientation, the angle θ between the magnetic moment and the magnetic field is 90 degrees.

The magnetic moment can be calculated as:

m = I * A = I * π * r^2

Substituting the values, we get:

m = 2 A * π * (0.20 m)^2 = 0.8π A m^2

Now, we can calculate the potential energy:

U1 = -m * B * cos(θ) = -0.8π * 0.5 T * cos(90°) = 0

2. Orientation 2: When the magnetic moment of the loop is parallel to the magnetic field.

In this orientation, the angle θ between the magnetic moment and the magnetic field is 0 degrees.

The magnetic moment remains the same as in Orientation 1:

m = I * A = I * π * r^2 = 0.8π A m^2

Now, we can calculate the potential energy:

U2 = -m * B * cos(θ) = -0.8π * 0.5 T * cos(0°) = -0.8π * 0.5 T * 1 = -0.4π T

The potential energy difference (ΔU) between Orientation 2 and Orientation 1 is given by:

ΔU = U2 - U1 = (-0.4π T) - 0 = -0.4π T

Now, let's calculate the value in Joules by multiplying with the current:

ΔU (Joules) = ΔU * I = (-0.4π T) * 2 A = -0.8π A T

So, the largest potential energy difference between the two orientations of the loop is -0.8π A T (Joules).

To find the largest potential energy difference between two orientations of the loop relative to the magnetic field, we need to consider the maximum change in the orientation of the loop with respect to the magnetic field. Let's break down the solution step by step.

Step 1: Calculate the magnetic moment of the current loop.
The magnetic moment of a current loop can be calculated using the formula:

μ = I * A,

where I is the current flowing through the loop and A is the area of the loop. The current is given as 2 A, and the area of the loop can be calculated using the formula for the area of a circle:

A = π * r^2,

where r is the radius of the loop given as 20 cm (0.2 m). Substitute the given values into the formula:

A = π * (0.2)^2 = 0.04π m^2.

Now, calculate the magnetic moment:

μ = 2 A * 0.04π m^2 = 0.08π A.m^2.

Step 2: Calculate the potential energy difference.
The potential energy of a magnetic dipole in a magnetic field is given by the formula:

U = -μ * B * cos(θ),

where U is the potential energy, μ is the magnetic moment, B is the magnetic field strength, and θ is the angle between the magnetic moment vector and the magnetic field vector.

Since we want to find the largest potential energy difference, we need to consider the extremes of the cosine function, which occur when θ is equal to 0° and 180°.

For θ = 0°:
U1 = -μ * B * cos(0°) = -μ * B * 1 = -μ * B.

For θ = 180°:
U2 = -μ * B * cos(180°) = -μ * B * (-1) = μ * B.

The potential energy difference between these two orientations is given by:
ΔU = U2 - U1 = μ * B - (-μ * B) = 2μ * B.

Substitute the values of μ = 0.08π A.m^2 and B = 0.5 T into the equation:

ΔU = 2 * 0.08π A.m^2 * 0.5 T = 0.08π A.m^2 * T.

Finally, to express the potential energy difference in Joules, we need to recall that 1 T·m^2 = 1 J, so:

ΔU = 0.08π A.m^2 * T = 0.08π J.

Therefore, the largest potential energy difference you can find between two orientations of the loop relative to the magnetic field is approximately 0.08π Joules.