Question

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

Answer 1

Hint Use Biot Savart Law to solve this question . For maximum value of the magnetic field try thinking of the conditions in the formula of biot savart law which can make the magnetic field largest.

Complete step by step solution:
Current: The rate of flow of electric charges is known as electric current.
Magnetic force: The force of attraction /repulsion which arises between different electrons by virtue of their motion.
Magnetic field: It is the field that describes the influence of magnetism on moving charges, electric current and magnetised material.
When a current carrying conductor is placed in a magnetic field, it experiences a magnetic force.
The value of this magnetic force is given by Biot Savart Law.
Biot Savart Law: It is an equation that gives the magnetic field by a current carrying wire.
It states that
1. Magnetic field is directly proportional to the small length segment of the wire.
2. Magnetic field is directly proportional to the current carried by the wire
3. Magnetic field is inversely proportional to the distance from any point P.
It is given by
$B = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{\vec i.d\vec l}}{{{r^2}}}$
Since we see a dot product in the above equation we need to solve it.
$ \Rightarrow B = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{idl\sin \theta }}{{{r^2}}}$
So we see here that
$B \propto i$
$b \propto dl$
$B \propto \dfrac{1}{{{r^2}}}$
$B \propto \sin \theta $
Since we see that the value of the magnetic field is directly proportional to the value of angle made between the length of the wire and the position of the point P.
Since in the formula all values are fixed,
The only variable term is $\sin \theta $.
$\sin \theta $ can be different depending on the position of point P.
We know that the maximum value of $\sin \theta $ can be 1, which is corresponding to the angle $90^\circ $.
Thus when the point P is perpendicular to the current length segment.
And thus max magnetic field will be obtained corresponding to $\sin \theta = 90^\circ $.

Note
This point should be remembered by the students that the maximum magnetic field will be obtained corresponding to the angle $\sin \theta = {90^0}$.