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The force between two magnetic poles is F when they are separated by a certain distance. If the distance between the two poles is doubled and the strength of each pole is tripled. The percentage change in the force between the two poles is:
A) 125%
B) 100%
C) 50%
D) 25%

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Last updated date: 14th Jun 2024
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Answer
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Hint The force between two magnetic poles is proportional to the product of the strength of the magnetic poles and inversely proportional to the square of the distance between these two magnetic poles. We will use this relation to determine the percentage increase in the poles when the pole strength is tripled and the distance between the poles is doubled.
Formula used: In this question, we will use the following formula
$\Rightarrow F = \dfrac{{{\mu _0}{M_1}{M_2}}}{{4\pi {d^2}}}$ where $F$ is the force between two magnetic poles of strength ${M_1}$ and ${M_2}$ at a distance $d$

Complete step by step answer
We know that the force between two magnetic poles of strength ${M_1}$ and ${M_2}$ at a distance $d$ can be calculated as:
$\Rightarrow F = \dfrac{{{\mu _0}{M_1}{M_2}}}{{4\pi {d^2}}}$
Now, we’ve been told that the distance between the two poles is doubled i.e. $d' = 3d$ and the strength of each pole is tripled, which implies${M_1}' = 3{M_1}$ and ${M_2}' = 3{M_2}$
So, the new force between these two poles will be
$\Rightarrow F' = \dfrac{{{\mu _0}{M_1}'{M_2}'}}{{d{'^2}}}$
On substituting $d' = 3d$, ${M_1}' = 3{M_1}$ and ${M_2}' = 3{M_2}$, we get,
$\Rightarrow F' = \dfrac{9}{4}\dfrac{{{\mu _0}{M_1}{M_2}}}{{4\pi {d^2}}}$
$\Rightarrow F' = \dfrac{9}{4}F$
The percentage increase in the force can be calculated as,
$\Rightarrow \% = \dfrac{{{\text{New force - old force}}}}{{{\text{Old force}}}} \times 100$
$\Rightarrow \% = \dfrac{{\dfrac{9}{4}F - F}}{F} \times 100$
On dividing the numerator and denominator by $F$, we get
$\% = 125\% $ which corresponds to option (A).

Note
Here we have assumed that the poles are in a vacuum which means there is no medium between them and there are no other magnetic poles in their vicinity that will affect the force they experience. We must be careful since we’ve been asked to find the percentage of the change in the force as compared to the old scenario and not the ratio of the two forces. Since the force is increased by 125%, it is more than doubled in the new scenario.