Question

Calculate the magnitude of the magnetic field at point Pof Fig E28.35. in terms of$R,l_1$and$l_2$. What does your expression give when $I_1=I_2$?

Short answer

1. The magnetic field at point p is$\frac{{\mu }_0\left|I_1-I_2\right|}{4R}$ .

2. The magnetic field at point p is zero if$I_1=I_2$.

Step-by-step solution

The magnetic field due to the current-carrying element

The magnetic field due to the current element at a particular point is given by Biot-Savart’s law.

According to Biot-Savart law magnetic field at a particular point is given by:

$\begin{array}{r}\overrightarrow{dB}=\frac{{\mu }_0}{4\pi }\frac{l\overrightarrow{d}/\times \widehat{r}}{r^2}\\ dB=\frac{{\mu }_0}{4\pi }\frac{Id/\sin \theta }{r^3}\end{array}$

Where dB is the magnetic field due to the current-carrying element,${\mu }_0$ is the permeability of the vaccum, l is the current through the conducting element, r is the distance from a point to the current-carrying element, dl is the differential length of the current element, and$\theta$ is the angle between$\overrightarrow{dl}$ and$\overrightarrow{r}$ .

The magnetic field at the center of a semi-circular current-carrying loop:

The magnetic field at the center of the semi-circular current-carrying loop is given by:

$B=\frac{{\mu }_{0}I}{4R}$

Where B is the magnetic field at the center of the semi-circular current-carrying loop, ${\mu }_0$is the permeability of the vaccum, l is the current through the wire, and R is the distance from the wire.

The direction of the magnetic field due to the current-carrying wire:

The direction of the magnetic field due to the current-carrying conductor can be given by the right-hand thumb rule.

According to the right-hand thumb rule, if thumb of the right-hand points along the direction of the current, then the remaining curled fingers of the same hand give the direction of the magnetic field due to the current.

Calculation of net magnetic field at point p

The magnetic field at point p due to parallel wires:

Here, the angles between differential lengths of left and right straight wires with point p are 0 and 180 degrees.

Therefore according to Biot-Savart law, the magnetic field at point p due to parallel wires is zero.

The magnetic field at point p (at the center of the semicircular loop) due to the semi-circular loop:

The magnetic field at point p (at the center of the semicircular loop) due to semi circular loop is:

$B_1=-\frac{{\mu }_0{I}_1}{4R}$

Here, according to the right-hand thumb rule, the direction of the magnetic field is clockwise. So conventionally $B_1$ is negative.

The magnetic field at point p (at the center of the semicircular loop) due semi-circular loop 2 is:

$B_2=\frac{{\mu }_0{I}_2}{4R}$

Here, according to the right-hand thumb rule, the direction of the magnetic field is anti-clockwise. So conventionally $B_2$ is positive.

So, the net magnetic field at point p is

_{$\begin{array}{r}{B}_{\text{net}}=B_2+B_1\\ B_{\text{net}}=\frac{{\mu }_0}{4R}\left|I_2-I_1\right|\end{array}$}

Thus, the net magnetic field at point p is $B_{\text{net}}=\frac{{\mu }_0}{4R}\left|I_2-I_1\right|$_.

Calculation of magnetic field at point p if the same current is flowing in both loops

Using expression:

$\begin{array}{r}{B}_{\text{net}}=B_2+B_1\\ B_{\text{net}}=\frac{{\mu }_0}{4R}\left|I_2-I_1\right|\end{array}$

If equal currents $\left(I_1=I_2\right)$is flowing through both wires, then the net magnetic field at point p is$B_{\text{net}}=0\mathrm{T}\text{.}$

Thus, the magnetic field at point p is zero if$I_1=I_2\text{.}$.