Question

The cross sections of five resistors which are made from the same metal and have the same lengths are shown below. Each of the resistors is connected to the same battery and an ammeter (see the circuit diagram below).
Resistors (a), (b) and (c) are solid and have radii of $R/2,R$ and $2R$ respectively.
Resistors (d) and (e) have a cylindrical hole in the center.
The outer radii of the resistors (d) and (e) are $R$ and $2R$.
The diameters of holes of these resistors are $R/2$ for the resistor (d) and $R$ for the resistor (e).
Rank the situations (a) through (e) from the lowest ammeter reading to the highest ammeter reading.

• A. e, d, b, c, a
• B. a, c, b, d, e
• C. c, e, b, a, d
• D. d, a, b, e, c
• E. c, e, b, d, a
• F. a, d, b, e, c

Answer 1

The correct option is F a, d, b, e, c

Conductance $K=\sigma \frac{A}{L}$ where $A$ is the cross-section area

$⟹$ $K\propto A$ $(\because \sigma$ and $L$ are constants $)$

(a) $A_a=\pi (R/2{)}^2=0.25$ $\pi R^2$

(b) $A_b=\pi (R{)}^2=\pi R^2$

(c) $A_c=\pi (2R{)}^2=4$ $\pi R^2$

(d) $A_d=\pi (R{)}^2-\pi (R/2{)}^2=0.75$ $\pi R^2$

(e) $A_e=\pi (2R{)}^2-\pi (R{)}^2=3$ $\pi R^2$

$\therefore$ $A_a<A_d<A_b>A_e<A_c$ $⟹$ $K_a<K_d<K_b<K_e<K_c$

Current in the resistors $I=VK$ $⟹I\propto K$

$⟹$ $I_a<I_d<I_b<I_e<I_c$ i.e option E is correct.