How Does Changing Electric Field Influence Magnetic Field in a Capacitor?

[GeorgeCostanz]

Homework Statement

A parallel-plate capacitor has circular plates with radius 49.0 cm and spacing 2.20 mm. A uniform electric field between the plates is changing at the rate of 1.90 MV/m/s. Find the magnitude of the magnetic field between the plates at a point 12.1 cm from the axis (entirely inside the capacitor).

Homework Equations

Ampere's Law for induced current

B(2∏r) = (μ-naught)(ε-naught)(A/d)(dV/dt)

r = .121m
μ-naught = 4∏x10^-7
ε-naught = 8.85x10&-12
d = .0022m
dV/dt = 1.9x10^6 V/(m/s)
A = ∏(.49m)^2

The Attempt at a Solution

the answer is B = 1.28x10^-12 T, but i can't seem to get that answer using my equation. i'd appreciate it if someone could direct me toward my error

thanks

 

[tiny-tim]

Hi George!

(try using the X² button just above the Reply box )

Show us your full calculations.

(In particular, what did you get for the current through the cylinder of radius 0.121 m ?)

 

[GeorgeCostanz]

@tiny-tim

sure.

I_d = [ (E_$\circ$A)/d ] * (dV/dt)

I_d = [ (E_$\circ$*(∏(.49²))/.0022 ] * (1.9x10⁶)

I_d = .005765 A

i guess

 

[tiny-tim]

B(2∏r) = (μ-naught)(ε-naught)(A/d)(dV/dt)

mmm … your formula seems to be correct, but I'm not getting the result of 1.28 10^-12 T either

 

[GeorgeCostanz]

i got the right answer using the following equation (googled the question)

B = [ (1/2)(r)(dV/dt) ] / C²

C = 3x10⁸ = speed of light in vacuum
dV/dt = 1.9x10⁶
r = .121m

not sure how the 2 equations are related tho

 

[tiny-tim]

ah! i took your word for it instead of looking at the original question …

A uniform electric field between the plates is changing at the rate of 1.90 MV/m/s.
…
dV/dt = 1.9x10^6 V/(m/s)

noooo … that wasn't dV/dt, it was dE/dt !

is everything clear now? ​

(and c² = 1/µ_oε_o, which btw would have have been a lot easier for you to use )

 

[GeorgeCostanz]

ah! i took your word for it instead of looking at the original question …


noooo … that wasn't dV/dt, it was dE/dt !

is everything clear now? ​

(and c² = 1/µ_oε_o, which btw would have have been a lot easier for you to use )

wow I'm slow
it's a miracle I've even made it this far

so for r < R,

B = (1/2)µ_oε_o(r)(dE/dt)

thus B = [ (1/2)(r)(dE/dt) ] / C²

guess i got mixed up with all these equations/derivations in front of me

thanks tiny-tim