Magnetic forces are required to study the force , a magnetic field exerts on charged particles, current elements and loops which is used in electrical devices in ammeters, volt meters, Galvano meters.

There are 3 ways in which force due to magnetic fields can be experienced.

- The force can be due to a moving charged particle in a Magnetic field.
- on a current element in an external B field.
- between two current elements.

**Force on a charged particle:-**

we know that .

.

where is the electric force on a stationary (or) moving electric charge in an electric field and is related to . where and are in the same direction.

a magnetic field can exert force only on a moving charge , suppose a charge Q is moving with velocity u (or) v in a magnetic field (B) is

.

from the equations is independent of velocity of the charge and performs work on the charge which changes its kinetic energy but depends on the charge velocity and is normal to it so work done it does not cause increase in the kinetic energy of the charge.

is small compared to except at high velocities.

so a charge which is in movement has both electric and magnetic fields.

Then .

.

This is known as Lorentz’s force equation. It relates mechanical force to electrical force.

if the mass of the charges particle is m

Then .

.

.

.

.

The solution to this equation is important in determining the motion of charged particles in in such cases the energy transfer is only by means of electric field.

**Force on a current element:-**

Consider a current carrying conductor , in order to find out the force acting on the current carrying element by the magnetic field .

.

.

.

.

is nothing but a elemental charge dQ moving with the velocity .

.

.

.

.

The line integral is for the current is along the closed path.

i.e,

The magnetic field produced by the current element does not exert force on the element itself just as a point charge does not exert force on itself.

So the magnetic field that exerts force on must be from the another element in other words the magnetic field is external to the current element .

Similarly we have force equations for other current elements and as follows

and .

So the magnetic field is defined as the force per unit current element

i.e, .(or) similar to .

so the describes the force properties of a magnetic field.

**Force between two current elements (Ampere’s force law):-**

Consider two current loops and then by Biot- Savart’s law both current elements produces respective magnetic fields so we may find the force on element s due to the field produced by .

Field produced by current element is is .

So the force applied on the element is by the field .

..

.

.

This is similar to coulomb’s law in electrostatics. Here it is law of force between two current elements and is analogous to coulomb’s law

.

Then the force acting on loop 2 due to the field produced by the current element t is nothing but

Note:- This is nothing but the Ampere’s force law that is the force between two current carrying conductors is given by it.

** **