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?

I tried the formula the answer 0.125J is wrong again. please tell me if there is another way to calculate the change in Potential energy?

To calculate the change in potential energy of a current loop in a magnetic field, you can use the equation:

ΔU = I * A * B * Δθ * sin(Δθ)

Where:
ΔU is the change in potential energy
I is the current
A is the area of the loop
B is the magnetic field strength
Δθ is the angle between the loop's normal and the magnetic field

In this case, the current is 2 A, the radius is 20 cm, and the magnetic field strength is 0.5 T.

First, we need to find the area of the loop. Since the loop is circular, its area can be calculated using the formula:

A = π * r^2

Where r is the radius of the loop. Plugging in the value, we get:

A = π * (0.2 m)^2
A = 0.04π m^2

Now, let's consider all possible orientations of the loop relative to the field to find the largest potential energy difference. To do this, we need to determine the maximum value of sin(Δθ). Since sin(Δθ) has a maximum value of 1, the maximum potential energy difference (ΔU) occurs when sin(Δθ) = 1.

Plugging in the values into the formula, we have:

ΔU = (2 A) * (0.04π m^2) * (0.5 T) * Δθ * 1

Simplifying, we get:

ΔU = 0.04π A T m^2 * Δθ

Now, we have to find the maximum angle, Δθ. The maximum angle occurs when the loop's normal is perpendicular to the magnetic field. In this case, the normal of the loop is the vector perpendicular to the circular plane of the loop.

The normal vector of a circle can be represented by a straight line that passes through the center of the circle. This line is also perpendicular to the plane of the circle. In this case, the normal vector is the same as the radius vector.

Therefore, the maximum angle occurs when Δθ = 90° or π/2 radians. Plugging this value into the equation, we get:

ΔU = 0.04π A T m^2 * (π/2)

Simplifying further, we have:

ΔU = 0.02π^2 A T m^2

Calculating the value, we get:

ΔU = 0.02 * (3.14159)^2 * 2 * 0.5
ΔU ≈ 0.6174 J

Therefore, the largest potential energy difference you can find between two orientations of the loop in the given magnetic field is approximately 0.6174 Joules.