(i) Express Biot-

(i) The vector form of Biot-Savart Law is given by

Hence the direction is perpendicular to dl and r and is given by the right hand corkscrew rule.

(ii) Let us consider a coil of radius R, carrying current I. We wish to find the magnetic field at point P as shown in the figure.

Here, distance AP is calculated using Pythagoras theorem.

The loop lies in the y-z plane and we wish to calculate the magnetic field along the x-axis. The Biot-savart law is

Consider an element dl on the loop and a position vector r which lies on the x-y plane. Hence, dl and r are perpendicular to each other (𝝓=90°).

This magnetic field can be resolved perpendicular to the x-axis and along the x-axis from elements all along the wire. As the perpendicular competes of mutually opposite elements are equal and opposite, the cancel out. So, we only have contributions from the horizontal components.

From the figure, we can see that

We know, Area,

And for coil of N turns, we have

(iii) The magnitude of magnetic field at the center is (d=0)

And at

Therefore, the ratio is

OR

(a) Moving charges experience a Lorentz force in a region of Electric and magnetic field. Consider a region where the electric and the magnetic fields are perpendicular to each other and the motion of the charged particle is perpendicular to both of them. This is illustrated in the figure given below:

In this situation, the Lorentz force is given by

When the electric and magnetic are varied such that their magnitudes become equal then,

Thus, only the charged particles of particular velocity go undeflected through the region. Hence, we can use crossed electric and magnetic field to pass only the charge particles of particular velocity

(b) A velocity selector or a velocity filter or a Wien filter is the electronic device which uses crossed electric and magnetic fields to get a stream of charged particles with identical velocities out of a mixture of particles having a wide range of velocities. The electric field deflects the electron upwards while the magnetic deflects them downwards. When the magnitudes of both are equal, the particles with velocity given by (1) will go undeflected.