Question

A magnetic field is given by B?=B0(x/x0)2k^, where B0 and x0 are constants

Part A

Find an expression for the magnetic flux through a square of side 2x0 that lies in the x-y plane with one corner at the origin and two sides coinciding with the positive x and y axes.

Express your answer in terms of the variables B0 and x0.

Answer 1

We know that flux(phi) = integral of B * dA

They give us the magnetic field, B = B_0( x/x_0)^2. They also tell us that the square lies in a coordinate system with its lower left corner at the origin, where the length of the square x = 2x_0. Thus dA of the square = 2x_0 dx.

Thus flux(phi) = integral of B_0( x/x_0)^2 * 2x_0 dx
where the limits are from 0 to 2x_0, from the origin to the length of the square.

distribute the square in B:
flux = integral of B_0( x^2/x_0^2) * 2x_0 dx

factor out the constants B_0, (1/x_0^2), and 2x_0:
flux = (B_0 * 2x_0)/x_0^2 integral x^2 dx

Evaluate at the limits:
(B_0 * 2x_0)/x_0^2) (x^3)/3 evaluated at [0 to 2x_0]

Fundamental Theorem of Calc:
(B_0 * 2x_0)/x_0^2) * (2x_0^3)/3

Cancel out like terms and simplify, thus our final answer is:

16/3*B_0*x_0^2