We can find the magnetic field around a wire on the axial
line of charge using the Biot-Servert’s law. We are not adding the derivation
here and the expression is as shown in the figure.

Let us consider a point on the axial line at a particular
distance and let us assume that we know the radius at any given point.

The expression for the magnetic field at any point can be expressed
as shown in the diagram. If we are measuring it at the center of the circle, we
need to equate the value to zero.

We can also find out the force acting basing on
Biot-Servert’s law. We can find the magnetic field at any given point using
this rule. We know that the magnetic induction is defined as the force
experienced by a unit north pole when placed in a magnetic field.

Thus we can measure the force as the product of the pole
strength and magnetic induction. As the field is small component, the force is
also small component. To get the total force acting on the point, we need to
integrate the given equation and we can get the total force as shown in the
diagram below.

This force will be maximum when the point is perpendicular to
the current carrying conductor. If the
angle is zero or 180 degree, the force will become zero as shown below.

To find the
direction of the force experienced
by the current carrying conductor
using Fleming Left hand rule. As per the law, if fore finger
indicate the direction of the magnetic
field and central finger indicates
the direction of the current then the thumb indicates the direction of the thrust or
force experienced by the current carrying conductor.

We can also measure
the force acting on a charge simply by
defining the current as the rate of
charge. We can define as the cross product of velocity of the charge and the
magnetic field and the product is multiplied
with the charge.

We can define the
unit of magnetic induction tesla basing on the above derivation. The magnetic field induction is the force experienced by
the conductor when a unit charge passing through a conductor with unit velocity at right angle.

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