Question

A counterclockwise current runs through a square wire loop in the xy plane centered at the origin. The length of each side is d. What is the magnetic field on the z axis?

Answer 1

Answer:

$B_{net} = \frac{2\sqrt2 \mu_0 i}{\pi d}$

Explanation:

Magnetic field due to straight current carrying wire is given by the formula

$B = \frac{\mu_0 i}{4\pi r}(sin\theta_1 + sin\theta_2)$

now we will have for one side of the square at its center position given as

$B = \frac{\mu_0 i}{4\pi (\frac{d}{2})}(sin45 + sin45)$

$B = \frac{2\sqrt2 \mu_0 i}{4 \pi d}$

now for the we have for complete square loop it will become 4 times of the one side

$B_{net} = 4 B$

$B_{net} = 4 \frac{2\sqrt2 \mu_0 i}{4 \pi d}$

$B_{net} = \frac{2\sqrt2 \mu_0 i}{\pi d}$