Question

A conductor ab of arbitrary shape carries current $I$flowing from b to a. The length vector $\overrightarrow {ab}$is oriented from a to b. The force $\overrightarrow F$ experienced by the conductor in a uniform magnetic field $\overrightarrow B$ is.
(A) $\overrightarrow F = - I(\overrightarrow {ab} \times B)$
(B) $\overrightarrow F = I(B \times \overrightarrow {ab} )$
(C) $\overrightarrow F = - I(\overrightarrow {ba} \times B)$
(D) All of these

Answer 1

We should know the formula for force on a current-carrying wire.
We should know the identity of$(A \times B) = - (B \times A)$.We should know the reversal of a vector $\overrightarrow {ab} = - \overrightarrow {ba}$.

Complete step by step answer:
A magnetic field is a vector quantity that describes the magnetic influence on moving electric charges, electric currents, and magnetized materials. A charge that is moving in a magnetic field experiences a force normal to its velocity and the magnetic field. Lorentzforce, the force exerted on a charged particle q moving with velocity v through an electric field E and magnetic field B. The entire electromagnetic force F on the charged particle is called the Lorentz force (after the Dutch physicist Hendrik A. Lorentz) and is given by
$F\; = \;qE\; + \;\left( {qv\; \times \;B} \right)$
Let us consider a wire of length ab, and a current is passing from b to a.
Then force experienced on the wire is,

We know that,
$dF = I(dl \times B)$
By integrating this we can find the value of F.
$\int {dF} = \int\limits_b^a {I(dl \times B)}$
$\int {dF} = I\int\limits_b^a {(dl \times B)}$
$F = I(ba \times B)$- - - - - - - - - - - - - - - - - - (1)
So Option (C) is correct.
We know that $\overrightarrow {ab} = - \overrightarrow {ba}$ . - - - - - - - - - - - - - - - - - - (2)
By rearranging (1) using (2)
We get,
$F = I(ba \times B) = - I(ab \times B)$- - - - - - - - - - - - - - - - - - (3)
Hence Option (A) is correct.
Again we know that $(A \times B) = - (B \times A)$- - - - - - - - - - - - - - - - - - (4)
By rearranging (3) using (4),
we get,
So,

F = - I(ab \times B) \\\ = - I( - B \times ab) \\\ = I(B \times ab) \\\ \end{gathered} $$ Since options (A), (B), (C) are correct, hence the correct Option is (D) **Note:** You should be aware of vector identities.If any two options are correct, then no need to find the other one since there is an option of all are correct. ( you can save time) You should also be aware and careful about sign conversion.