Question

Calculate the magnitude and direction of the magnetic field at point Pdue to the current in the semicircular section of wire. (Hint:Does the

current in the long, straight section of the wire produce any field at P?)

Short answer

The magnitude of the magnetic field at point p is$\frac{{\mu }_0}{4R}$ .

The direction of magnetic field into the page is into the page.

Step-by-step solution

Step 1:  The magnetic field due to current carrying element

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

According to Biot-Sarvant law magnetic field at particular point is given by

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

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

The magnetic field at center of semi circular current carrying loop

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

$B=\frac{{\mu }_0{l}_0}{4R}$

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

The direction of magnetic field due to current carrying wire

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

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

Direction of magnetic field at point p

According to right hand thumb rule direction of magnetic field at point p due to current carrying semi circular loop is inward the plane

Calculation of net magnetic field at point p

Magnetic field at point p due parallel wires

Here, the angles between differential lengths of left and right straight wires with point p aredegrees.

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

Magnetic field at point p (at center of semicircular loop) due semi circular loop

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

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

So, the net magnetic field at point p only due to semicircular loop is$B=\frac{{\mu }_{0}l}{4R}$ _.

Thus, The magnitude of the magnetic field at point p is $\frac{{\mu }_{0}l}{4R}$.