# Rank the paths according ∫B.ds around the paths during the discharging of the capacitor

hidemi
Homework Statement:
The diagram shows one plate of a parallel-plate capacitor from whin the capacitor. The plate is circular and has radius R. The dashed circles are four integration paths and radii of r1=R/4, r2=R/2, r3=3R/2, and r4=2R. Rank the paths according to the magnitude of ∫B.ds
around the paths during the discharging of the capacitor, least to greatest.

A) 1, 2 and 3 tie, then 4
B) 1,2,3,4
C) 1, then 2 and 4 tie, then 3
D) 4,3,1,2

E) all tie
Relevant Equations:
B = μ0*I / 2πr
My calculation is as attached. Where am I wrong?

#### Attachments

Homework Helper
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I agree with your results for B. So, it looks like (C) would be the correct answer if the question asked for the ranking of B on each path, rather than asking for the ranking of ∫Bds. None of the answers appear to be correct for the ranking of ∫Bds.

[Edit: For the ranking of ∫Bds, you can argue that one of the answers is correct if you take into account the fringing of the E field of the capacitor.]

Last edited:
• Delta2 and hidemi
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... Rank the paths according to the magnitude of ∫B.ds
.
My calculation is as attached. Where am I wrong?
From your working it looks like you have tried to rank the different values of B. But the question specifically says “Rank ... according to the magnitude of ∫B.ds”. You answered the wrong question!

The question is about displacement current. So ideally you shouldn’t use ##I##. Use (for example) ##I_D##.

(Explanatory note follows, if needed:

Inside of the capacitor is a dielectric (vacuum, or air, or some other insulator). This mean no physical current (moving charge) actually crosses the gap between the plates.

The changing electric flux inside the capacitor generates a magnetic field. ##I_D## is a ‘fictitious’ current which would create the same magnetic field as the changing electric flux. So, when finding the magnetic field, we can ‘pretend’ ##I_D## is actually flowing between the plates like a real current.)

Since the electric field is uniform, ##I_D##’s distribution is the same as a uniform current through a cylindrical conductor radius R. The current, ##I_r##, through an ‘inner cylinder’ (r≤R) is ##I_D \frac {r^2}{R^2}## because it is proportional to cross-sectional area.

The current, ##I_r##, through an ‘outer cylinder’ (r>R) is , ##I_r=I_D##, because it is the total current.

Ampere’s law tell us ##\int B.ds = \mu_0 I_r##. So we are simply being asked to rank the values of ##\mu_0 I_r## for different value of r.

Once that’s fully understood, no calculations at all are needed to answer the question! What do you think the answer should be?

• TSny, Delta2 and hidemi
hidemi
Thank you so much.

hidemi
I agree with your results for B. So, it looks like (C) would be the correct answer if the question asked for the ranking of B on each path, rather than asking for the ranking of ∫Bds. None of the answers appear to be correct for the ranking of ∫Bds.

[Edit: For the ranking of ∫Bds, you can argue that one of the answers is correct if you take into account the fringing of the E field of the capacitor.]
Thank so much.

Homework Helper
Gold Member
The correct answer for the ranking of the magnitude (absolute value) of ##\int \mathbf{B}\cdot d\mathbf{s}## (and not for the ranking of B as your work is) is 1<2<3=4.

• hidemi
hidemi
The correct answer for the ranking of the magnitude (absolute value) of ##\int \mathbf{B}\cdot d\mathbf{s}## (and not for the ranking of B as your work is) is 1<2<3=4.
Yes I agree with you, thanks for commenting!

• Delta2