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?

ΔPE=Work

W=∫τ•dφ=∫pBsinφ•dφ=
=∫IABsinφ•dφ=∫Iπr²Bsinφ•dφ=
=Iπr²Bcosφ (limits of integral are π/2 and 0)= Iπr²B

W=Iπr²B=2•π•0.2²•0.5=0.125 J

.125 J is wrong answer. can you check lease. thanx

To find the largest potential energy difference between two orientations of the current loop, we can use the formula for potential energy in a magnetic field.

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

PE = -magnetic moment · magnetic field · cosine(theta)

Here, the magnetic moment (μ) is given by the equation:

μ = current · area

In this case, the radius of the current loop is given as 20 cm or 0.2 meters, and the current passing through the loop is 2 A. The area of the loop can be calculated using the formula for the area of a circle:

Area = π · radius^2

Let's calculate the magnetic moment and area of the loop:

Area = π · (0.2)^2 = 0.04π square meters

μ = 2 A · 0.04π square meters = 0.08π A·m^2

Now, we need to calculate the potential energy for two different orientations of the loop:

For the first orientation, the angle between the magnetic field and the loop's plane is 0 degrees (θ = 0). The potential energy for this orientation can be calculated as:

PE1 = -μ · magnetic field · cosine(0)

Since cosine(0) = 1, the equation simplifies to:

PE1 = -μ · magnetic field

Substituting the values:

PE1 = -0.08π A·m^2 · 0.5 T = -0.04π Joules

For the second orientation, the angle between the magnetic field and the loop's plane is 180 degrees (θ = 180). The potential energy for this orientation can be calculated as:

PE2 = -μ · magnetic field · cosine(180)

Since cosine(180) = -1, the equation simplifies to:

PE2 = μ · magnetic field

Substituting the values:

PE2 = 0.08π A·m^2 · 0.5 T = 0.04π Joules

The potential energy difference between these two orientations can be calculated as:

ΔPE = PE2 - PE1

ΔPE = (0.04π) - (-0.04π) = 0.08π Joules

Therefore, the largest potential energy difference between two orientations of the current loop is 0.08π Joules.