A Gaussian surface in the shape of a right circular cylinder with end caps has a radius of 12.5 cm and a length of 112 cm. Through one end there is an inward magnetic flux of 36.4 μWb. At the other end there is a uniform magnetic field of 2.33 mT, normal to the surface and directed outward. What are the (a) magnitude and (b) direction (inward or outward) of the net magnetic flux through the curved surface?

A) Magnitude of flux; Φ_c = 77.97 μWb

B) Direction is inwards

Explanation:

A) The magnetic flux through a closed surface is zero. Thus, the summation of the flux through the two bases and the two bases and the curved surface equals zero and is written as:

Φ_s1 + Φ_s2 + Φ_c = 0

Where;

Φ_s1 is the flux through one end and it's given as; - 36.4 μWb

(negative sign denotes the direction which is inwards)

Φ_s2 is the flux through the other end is given by a formula;

Φ_s2 = BA

Area is; A = πr²

Thus; Φ_s2 = Bπr²

We are given;

B = 2.33 mT = 2.33 x 10^ (-3) T

r = 12.5 cm = 0.125 m

Thus,

Φ_s2 = 2.33 x 10^ (-3) x π (0.125²)

Φ_s2 = 114.37 x 10^ (-6) Wb

Thus;

Φ_s2 = 114.37 μWb

Since we know Φ_s1 and Φ_s2, let's plug them into the first equation to find the magnetic flux; Φ_c

Thus;

-36.4 μWb + 114.37 μWb + Φ_c = 0

Φ_c = - 77.97 μWb

The magnitude is the absolute value which is; Φ_c = 77.97 μWb

B) Since the flux has a negative value, thus the direction is inwards