Question

What is the value of the magnetic field at the center of a square wire of length a on the side?

Answer 1

see the image for answer

Answer 2

Concept:

A magnetic field, which is a type of vector field, describes the magnetic influence on moving electric charges, electric currents, and magnetic materials.

Given:

A square wire with each side of length a.

Find:

The magnetic field at the center of the square wire.

Solution:

Let the current in the square wire be I.

The distance from the center of the square to the midpoint of the wire of one side of the wire to the center is d.

Therefore, $d=\frac{a}{2}$

Now, the magnetic field at the center is:

$B=\frac{\mu_{0}i}{4\pi d} [sin \alpha + sin\beta]$

Now, the vertices of the square have an angle of 90 degrees.

Therefore, α = β = 45°

So,

$B=\frac{\mu_{0}i}{4\pi d} [\frac{1}{\sqrt{2} } + \frac{1}{\sqrt{2} }]$

$B=\frac{\mu_{0}i}{2\sqrt{2} \pi d}$

$B=\frac{\mu_{0}i}{2\sqrt{2} \pi \frac{a}{2} }$

$B=\frac{\mu_{0}i}{\sqrt{2} \pi a}$

So,  the total magnetic field at the center of the square wire is:

$B=(4)\frac{\mu_{0}i}{\sqrt{2} \pi a}$

$B=\frac{4\sqrt{2} \mu_{0}i}{ \pi a}$

The magnetic field at the center of a square wire is $B=\frac{4\sqrt{2} \mu_{0}i}{ \pi a}$.

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