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Q.

What happens to resistance if the diameter of the wire becomes half?

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a

Increases

4 times

b

Increases

2 times

c

Increases

6 times

d

None

answer is A.

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Detailed Solution

The resistance increases four times if the diameter of the wire becomes half.
When we modify the area of the cross-section of a wire, its length varies while its volume remains constant. Let A and L be the initial cross-sectional area and length of the wire and R be the resistance of a wire. Then the resistance of the wire can be calculated by using a formula.

R = ρ \frac{L}{A}

………………………… (i)
Now if the cross-sectional area of the wire is half, then the volume of wire

= AL

V = \frac{A}{2} \times L^{'}

{[L}^{'}

=represents the new length;

V = volume of wire

]
Given,

L^{'} = 2 L

So, new resistance

R^{'}

will become

R^{'} = \frac{ρL'}{2 A}

R^{'} = \frac{ρ 2 L}{\frac{A}{2}}

R^{'} = \frac{ρ 2 L \times 2}{A}

R^{'} = \frac{4 ρL}{A}

………………… (ii)
On comparing equation (i) and (ii) we get,

R^{'} = 4 R

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