CBSE Class 12 · 2019 · Magnetic Effects of CurrentLong Answer

Two infinitely long straight wires

A_{1}

and

A_{2}

carrying currents

I

and

2 I

flowing in the same direction are kept a distance '

d

' apart. Where should a third straight wire

A_{3}

carrying current

1.5 I

be placed between

A_{1}

and

A_{2}

so that it experiences no net force due to

A_{1}

and

A_{2}

? Does the net force acting on

A_{3}

depend on the current flowing through it?

Solution

Let

A_{3}

be at distance

x

from

A_{1}

, so it is

(d - x)

from

A_{2}

. The force per unit length from a long wire is

\frac{μ _{0} I _{a} I _{b}}{2 π r}

.
For zero net force the two forces (which are opposite, since

A_{1}, A_{2}

are on opposite sides) must be equal in magnitude:

\frac{μ _{0} ( 1.5 I ) I}{2 π x} = \frac{μ _{0} ( 1.5 I ) ( 2 I )}{2 π ( d - x )} .

\frac{1}{x} = \frac{2}{d - x} \Rightarrow d - x = 2 x \Rightarrow x = \frac{d}{3} .

A_{3}

is placed at

\frac{d}{3}

from

A_{1}

(the weaker-current wire).
The net force magnitude is

F = I_{3} (B_{1} + B_{2})

per unit length, which is proportional to

I_{3}

; the null position itself is independent of

I_{3}

, but away from it the force does depend on the current in

A_{3}

x = \frac{d}{3} from A_{1}; net force \propto I_{3}

CBSE Class 12 Board · 2019 · Subjective