Question

A wire with current$\mathbf{i}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{00}\mathbf{}\mathbf{A}$is shown in Figure. Two semi-infinite straight sections, both tangent to the same circle, are connected by a circular arc that has a central angle $\mathit{\theta }$and runs along the circumference of the circle. The arc and the two straight sections all lie in the same plane. If $\mathbf{B}\mathbf{=}\mathbf{0}$at the circle’s center, what is $\mathit{\theta }$?

Short answer

The value of $\varnothing$for the zero magnetic field at the circle’s center is$\theta =2.00\text{rad}$.

Step-by-step solution

Given

1. The current flowing through the wire is $i=3.00\mathrm{A}$
2. The magnetic field at the circle’s center is $B=0\mathrm{T}$.

Determine the formulas for the magnetic field as:

Formula:

$B_{straight}=\frac{{\mu }_{0}i}{4\pi R}$

$B_{arc}=\frac{{\mu }_{0}i\varnothing }{4\pi R}$

Calculate the value of ∅  for the zero magnetic field at the circle’s center

The magnetic field due to a current in semi-infinite straight wire is as follows:

$B_{straight}=\frac{{\mu }_{0}i}{4\pi R}$

According to the right hand rule, both wires produce a magnetic field that is pointing out of the page.

The magnetic field due to the current in a circular arc of the wire is:

$B_{arc}=\frac{{\mu }_{0}i\varnothing }{4\pi R}$

According to the right hand rule, it is pointing into the page.

The total magnetic field for the system is:

$B=2B_{straight}=B_{arc}$

$B=2\left(\frac{{\mu }_{0}i}{4\pi R}\right)-\frac{{\mu }_{0}i\varnothing }{4\pi R}$

$B=2\left(\frac{{\mu }_{0}i}{4\pi R}\right)-\frac{{\mu }_{0}i\varnothing }{4\pi R}$

$0T=2\left(\frac{{\mu }_{0}i}{4\pi R}\right)-\frac{{\mu }_{0}i\varnothing }{4\pi R}$

Solve further as:

$2\left(\frac{{\mu }_{0}i}{4\pi R}\right)=\frac{{\mu }_{0}i\varnothing }{4\pi R}$

$\varnothing =2.00\text{rad}$