Physics quiz question: wide versus narrow bar

Physics 205A Quiz 7, fall semester 2012
Cuesta College, San Luis Obispo, CA

Cf. Giambattista/Richardson/Richardson, Physics, 2/e, Problem 14.49

A steel bar has twice the cross-sectional area as another steel bar of the same length. The bottom of each bar is immersed in 0° C water, while the top of each bar is heated to 75° C. (Ignore the very slight thermal expansion of these bars). The wider bar has ________ the thermal resistance of the narrower bar.
(A) one-fourth.
(B) one-half.
(C) twice.
(D) four times.
(E) (There is a tie.)

Correct answer: (B)

The thermal resistance R is given by:

R = d/(Κ·A),

where d corresponds to the (identical) length of these bars, Κ is the (identical) thermal conductivity of these bars, and A is the cross-sectional area of the bars. With A_wide = 2·A_narrow, then for these two bars:

R_wide = d/(Κ·A_wide),

R_narrow = d/(Κ·A_narrow),

The wider bar will have one-half the thermal resistance of the narrower bar:

R_wide = d/(Κ·A_wide) = d/(Κ·(2·A_narrow)) = (1/2)·R_narrow.

Sections 70854, 70855
Exam code: quiz07Di5k
(A) : 0 students
(B) : 24 students
(C) : 18 students
(D) : 3 students
(E) : 5 students

Success level: 48%
Discrimination index (Aubrecht & Aubrecht, 1983): 0.50