Find the flux given the field.

Question

A magnetic field is given by B(x,y,z) = C(xi + yj − 2zk) where C is a constant. Find the flux of B across the surface of a square coil of side length a when it is placed a distance 2a from the origin with its centre on the z axis and its sides parallel to the x and y axes.

 

attempt: the flux is the integral of the dot product of the field with a small area element (da). However I cannot imagine how the coil can be parallel to both x and y axes? Also what is the importance of knowing the distance from the origin when the length of each side is given?

Answer 1

The area is a square perpendicular to the z-axis, so it lies in the x-y plane.  Therefore, one side can be parallel to the x-axis, and another can be parallel to the y-axis.  Based on the given information, it looks like the area vector is 

 

A = a²n = a²k

 

dA = dxdy

 

Φ = ∫∫B·ndA = ∫∫B·ndxdy

 

B·n = B·k = B_z = -2Cz

 

Φ = ∫₀^a∫₀^a(-2Cz)dxdy = -2Cza²

 

But what is z?  Here is where we use the information that the square is placed a distance 2a from the origin.  Since the square is located along the z-axis,

 

z = 2a = constant ⇒

 

Φ = -2Cza² = -2C(2a)a² = -4Ca³