Answer:
The current in the loop is [tex]I = \frac{W}{2 \pi R^2 B}[/tex]
Explanation:
From the question we are told that
The magnetic field(B) points in a positive z direction so mathematically
[tex]B = 0\r i + 0\r j + B\r k[/tex]
[tex]= B\r k[/tex]
The radius is given as R
Let denote the constant unknown current as [tex]I[/tex]
Generally the magnetic dipole moment of the field is mathematically represented as
[tex]\mu = IA[/tex]
And its direction is the positive z-direction So
[tex]\mu = IA \r k[/tex]
[tex]A = \pi R^2[/tex]
The potential energy is mathematically represented as
[tex]V = - \mu B cos \theta[/tex]
The negative sign show that it increases when work is done against the field
For the loop the initial potential energy is
[tex]V_i = -\mu B cos (0)[/tex]
[tex]=-I \pi R^2 B[/tex]
The final potential energy is
[tex]V_f = -\mu B cos(180)[/tex] the 180° showing the loop is flipped over
[tex]=I \pi R^2 B[/tex]
The change in the Potential energy give the workdone
Let denote this workdone with W So,
[tex]W = V_f -V_i[/tex]
[tex]= I \pi R^2 B - (-I \pi R^2 B)[/tex]
[tex]= 2 \pi I R^2 B[/tex]
Making the current subject of the formula we have
[tex]I = \frac{W}{2 \pi R^2 B}[/tex]
