Question

A long current carrying conductor of length $l$ is placed in a uniform magnetic field strength $B.$ If current in the conductor is $i$ $A$, write down the formula of force exerted on the current carrying conductor. What will be the maximum force? Write its direction.

Answer 1

Hint: Use the formula for force on a current carrying conductor in a magnetic field. Then use the concept of range to find the concept of range to find the maximum value of force.
$\overrightarrow F = \overrightarrow I \times \overrightarrow B l$

Complete step by step answer:
We know that, the force on the current carrying conductor of length $l$in a uniform magnetic field $B$is given
$\overrightarrow F = \overrightarrow I \times \overrightarrow B l$
Where, $I$is current
We have $I = iA$
Therefore, $\overrightarrow F = iBl\sin \theta \widehat n$
$\left( {\because \overline a \times \overline b = \left| {\overline a } \right|\left| {\overline b } \right|\sin \theta \widehat n} \right)$
$ \Rightarrow \left| {\overrightarrow F } \right| = iBl\sin \theta \left( {\because \left| {F = 1} \right|} \right)$
Since, $i,B$ and $l$ are constant, the value of force will depend on the variation in $\sin \theta $
Therefore, force is maximum when $\sin \theta $ is maximum.
We know that $\left| { \leqslant \sin \theta \leqslant 1} \right|$ i.e. the maximum value of $\sin \theta $ = $1$ and $\sin \theta = 1$
Therefore, the maximum value of force will be${F_{\max }} = lBl$
and it will be maximum.
When the angle between current and magnetic field is $90^\circ $
In $\overline a \times \overline b ,$
The direction of $\overline a \times \overline b $ is perpendicular to the p line.
Containing $\overline A $ and $\overline b $
Therefore, the direction of force will be perpendicular to the plane containing current and magnetic field.

Note:
You should know the range of basic functions to simplify the question. We didn’t know the range of $\sin \theta, $ then we would have to use differentiation to find the maximum value of the force, which would have wasted time.