Question

Two identical cycle wheels (geometrically) have a different number of spokes connected from centre to rim. One has 20 spokes and the other having only 10(the rim and the spokes are resistance-less). One resistance of value R is connected between centre and rim. The current in R will be:
(A) Double in first wheel than in the second wheel
(B) Four times in the first wheel than in the second wheel
(C) Will be double in the second wheel than that of the first
(D) Will be equal in both the wheels

Answer 1

Hint
To find the current in the wheels, we need to find out the equivalent resistance of both the wheels. Then, we need to compare the currents in the wheels to get a result.
${R_{eq}} = \dfrac{R}{n}$
$I = \dfrac{V}{{{R_{eq}}}}$
Where, R is the resistance, V is voltage across the ends, I is the current.

Complete step by step answer
Let us consider the resistance of the spokes of the wheels to be R, which is equal for both the wheels.
Now, there are 20 spokes in wheel 1 and they are connected in parallel, then the equivalent resistance of the spokes becomes,
${R_{eq}} = \dfrac{R}{n}$
$ \Rightarrow {R_{eq}} = \dfrac{R}{{20}}$
Now, there are 10 spokes in wheel 2 and they are connected in parallel, then the equivalent resistance of the spokes becomes,
$ \Rightarrow {R_{eq}} = \dfrac{R}{{10}}$
Now, the current in the first wheel is,
$I = \dfrac{V}{{{R_{eq}}}}$
$ \Rightarrow I = \dfrac{V}{{\dfrac{R}{{20}}}}$
Then,
$ \Rightarrow \dfrac{I}{{20}} = \dfrac{V}{R}$
Now, the current in the second wheel is,
$I = \dfrac{V}{{{R_{eq}}}}$
$ \Rightarrow I = \dfrac{V}{{\dfrac{R}{{10}}}}$
Then,
$ \Rightarrow \dfrac{I}{{10}} = \dfrac{V}{R}$
Thus, when we compare both the currents, the current in R for both the wheels is equal as we get an equivalence relation above. Thus, the currents are equal.
Hence, the correct answer is option (D).

Note
It is given in the question to compare the current through the resistance R in both wheels. We need to consider the current in the resistance and do not compare the current in the wheels. The current in the wheels might differ, but the current through the resistance is the same.