Equal but opposite currents I and I flow in two long paralle

Equal but opposite currents +I and -I flow in two long parallel plates. The plates have width w and seperation d, where d is small.

a) Neglecting the edge effects, find the magnetic field between the plates from Ampere\'s Law.

Solution

We can consider the long plates as thin long wires carrying current +I and –I placed very closely

The magnetic field at point between the plates is sum of the fields due to the wires.

consider a point at a point at the middle from the two plates

using Ampere’s law draw a circle of radius d/2 with center from a point on each plate

B = ₀I/2(d/2) + ₀I/2(d/2) = 2 ₀I/d

at distance x away from the mid point the field will be

B = ₀I/2{1/(d/2+x) +1/(d/2-x)}

as d is small and for small values of x we can approximate it as

B 2 ₀I/d

The magnetic field along the width of the plates left to right or right to left depending on the direction of the currents.

other components of the two fields cancel up each other

Around the middle of the two plates there is uniform magnetic filed

B = 2 ₀I/d