Question

The cross sections of five resistors which are made from the same metal and have the same lengths are shown 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 these resistors from most conductive to least conductive.

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

Answer 1

Answer: (D)

c, e, b, d, a

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

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

Hence option D is correct.