Question

A system consists of two parallel planes carrying currents producing a uniform magnetic field of induction $B$ between the planes. Outside this space there is no magnetic field. If the magnetic force acting per unit area of each plane is ${F_1} = \dfrac{{{B^2}}}{{x{\mu _0}}}$. Find $x$.

Answer 1

Use the formula for the circulation law. Also use the formula for magnetic force due to the magnetic field due to the current carrying wire. First determine the net magnetic field due to each plane. Then calculate the magnetic force per unit area exerted by the one plane on the other plane and determine the value of $x$.

Formulae used:
The expression for circulation law is
$B = {\mu _0}i$ …… (1)
Here, $B$ is the magnetic field, ${\mu _0}$ is permeability of free space and $i$ is current per unit length of the conductor.
The magnetic force $F$ due to the magnetic field in the current carrying wire is
$F = Bil$ …… (2)
Here, $B$ is the magnetic field, $i$ is the current in the conductor and $l$ is the length of the conductor.

Complete step by step answer:
We have given that the magnetic field due to the currents flowing in the two planes is $B$ and there is no magnetic field in the space outside the two planes.From this, we can conclude that the magnetic field due to the currents flowing in the two planes outside the two planes cancel each other.This shows that the direction of the electric current in the two planes is opposite to each other.
Rearrange equation (1) for $i$.
$i = \dfrac{B}{{{\mu _0}}}$
Hence, the magnetic field due to one plane is $\dfrac{B}{2}$ which is half of the total magnetic field.

The force due to one plane on the other plate is given by
$F = \dfrac{B}{2}il$
Here, $i$ is the current per unit length.
$\Rightarrow F = \dfrac{B}{2}i\left( {{\text{length}}} \right)\left( {{\text{breadth}}} \right)$
Substitute $\dfrac{B}{{{\mu _0}}}$ for $i$ in the above equation.
$\Rightarrow F = \dfrac{B}{2}\left( {\dfrac{B}{{{\mu _0}}}} \right)A$
$\therefore \dfrac{F}{A} = \dfrac{{{B^2}}}{{2{\mu _0}}}$

Hence, the value of $x$ is 2.

Note: The students should keep in mind that we have replaced the length in the formula for the magnetic force by the breadth as the plane has length as well as breadth. One can also solve the same question by another method. One can use the Ampere’s law instead of the circulation law to determine the value of the current per unit length in terms of the magnetic field and permeability of free space.