Question

Figure shows four identical currents i and an
amperian loops encircling them. We shall calculate
∆B.dl in the direction marked. The correct value
is​

Answer 1

Given:

Figure shows four identical currents i and an

amperian loops encircling them.

To find:

Value of $\oint \vec{B}.d\vec{l}$

(Options given in image).

Calculation:

Consider upward direction to be +ve.

Starting from left to right:

1st loop:

$\displaystyle \oint \vec{B} \: . \: d \vec{l} = - \mu_{0}i \: \: \: \: ......(1)$

2nd loop:

$\displaystyle \oint \vec{B} \: . \: d \vec{l} = \mu_{0}i \: \: \: \: ......(2)$

3rd loop:

$\displaystyle \oint \vec{B} \: . \: d \vec{l} = \mu_{0}(2i) \: \: \: \: \: ......(3)$

So, net magnetic circulation:

$\displaystyle \oint \vec{B} \: . \: d \vec{l} = \mu_{0}(2i + i - i)$

$= > \displaystyle \oint \vec{B} \: . \: d \vec{l} = \mu_{0}(2i)$

$= > \displaystyle \oint \vec{B} \: . \: d \vec{l} = 2\mu_{0}i$

So, final answer is:

$\displaystyle \boxed{ \bf{ \oint \vec{B} \: . \: d \vec{l} = 2\mu_{0}i}}$