Question

When is the force experienced by a current-carrying conductor placed in a magnetic field largest?

Answer 1

Hint
Here we have to consider a current-carrying conductor kept in a magnetic field. The charges constituting the electric current experience a Lorentz force. Hence the current-carrying conductor kept in a magnetic field will be deflected. When the charged particle moves perpendicular to the magnetic field, the magnetic Lorentz force will be maximum.

Complete step by step answer
We are considering the current-carrying conductor placed in a magnetic field. Let the length of the conductor be${\text{ }}l{\text{ }}$. The conductor has a uniform area of the cross-section ${\text{ }}A.{\text{ }}$The conductor is kept in a magnetic field ${\text{ }}B{\text{ }}$at an angle${\text{ }}\theta {\text{ }}$, let the number density of electron in the conductor be ${\text{ n}}{\text{. }}$
Now, we can write the number of free electrons in the conductor as,
Total free charge${\text{ }} = nalq$
We have a current carrying conductor kept in a magnetic field. The magnetic Lorentz force can be written as,
$\Rightarrow {\text{ }}F{\text{ }} = {\text{ }}qvB{\text{ }}\sin \theta $
Let ${\text{ }}v{\text{ }}$be the drift velocity of the charges, then the Lorentz force acting on the charges can be written as
$\Rightarrow {\text{ }}nAlq(v \times B){\text{ }} = {\text{ }}nAlqvB{\text{ }}\sin \theta {\text{.}}$
But, we know that ${\text{ }}nAvq = I$
Where${\text{ }}I{\text{ }}$ is equal to the current flowing through the conductor.
Therefore, now we can say that the force acting on the conductor
$\Rightarrow {\text{ }}F = BIl\sin \theta = I\left( {\vec l \times \vec B} \right)$
This is the force acting on a charge carrying conductor of length${\text{ }}l{\text{ }}$ kept in a magnetic field ${\text{ }}B{\text{ }}$
When ${\text{ }}\theta = {90^ \circ }$
That is when the conductor and the magnetic field are perpendicular in direction.
$\Rightarrow \sin \theta = \sin {90^ \circ } = 1$
Then, the force is,${\text{ }}F = BIl,{\text{ }}$ which is maximum.
This means that the force is maximum when the conductor is perpendicular to the magnetic field.

Note
The force will be minimum i.e. ${\text{ }}F = 0{\text{ }}$when
$\sin \theta = 0$ i.e. when ${\text{ }}\theta = 0{\text{ }}$
This means that the magnetic field will be minimum (zero) when the conductor is parallel to the magnetic field and \theta = 180$
This means that the magnetic field will again be minimum (zero) when the conductor is kept opposite to the direction of the magnetic field.