Question

(a) Define one tesla.
(b) Derive an expression for force experienced by a current carrying straight conductor placed in a magnetic field. How can we find the direction of force?

Answer 1

Hint: When charge is at rest only electric field id produce but when charge is under motion then both electric field and magnetic field is produced. Now if there is an existing magnetic field and if a charge is under motion in that existing magnetic field then the magnetic force will be acting on that charge and that force expression will be base for the following derivation.
Formula used:
$\eqalign{
  & F = qvB\sin \theta \cr
  & F = i\left( {\mathop l\limits^ \to \times \mathop B\limits^ \to } \right) \cr
  & v = \dfrac{l}{t},l = vt \cr
  & i = \dfrac{q}{t},q = it \cr} $

Complete answer:
Let us assume that a charge q is moving with velocity ‘v’ in a magnetic field ‘B’ and the angle between that velocity vector and magnetic field vector is ϴ. Due to that magnetic field there will be a force acting on that moving charge which is $F = qvB\sin \theta $ where q is the magnitude of charge v is the velocity of that charge and B is the magnitude of that magnetic field. Now one has to remember that this force can change only direction of velocity but not magnitude of velocity.
We have $q = it$. By substituting this in force expression we get
$\eqalign{
  & F = itvB\sin \theta \cr
  & l = vt \cr
  & F = ilB\sin \theta \cr
  & F = i\left( {\mathop l\limits^ \to \times \mathop B\limits^ \to } \right) \cr} $
Here l is the length of the conductor
From the above expression we can tell that the force is perpendicular to the plane containing both length vector and the magnetic field vector.
The direction of force can be obtained by Fleming's right hand rule. Putting the palm in the direction of current and curling it along the direction of the magnetic field. Then the direction of the thumb shows the direction of force.
In that expression if 1A current is passing through a wire of 1m length and if there is a magnetic field B which produces 1N of force on it then the intensity of that magnetic field would be one tesla.

Note:
In the expression where magnetic force is acting on a charge, that force cannot change the direction of velocity because that force will be in perpendicular direction to the velocity which means work done by that force is zero which can’t change the kinetic energy of that charge. Same applies with force on current carrying wire too as it can’t change the velocity of the wire.