Magnetic Field in Capacitor

[Ben Whelan]

Homework Statement

A cylindrical parallel plate capacitor of radius R is discharged by an external current I. The
total electric field flux inside the capacitor changes at a rate dΦ_e/dt = I/ ε₀. What is the
strength of the resulting magnetic field B(r, I) inside the capacitor at a radial distance r
from the centre axis? Start the answer with Maxwell’s extension to Ampere’s law.
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Homework Equations

So using the line integral ∫B⋅dl= μ₀ (I + I_d )[/B]

The Attempt at a Solution

you get B⋅2πr= μ₀ (I + I_d )

However i don't understand what I and I_d are?

I have the equation I_d= ε₀ dΦ_e/dt

however i don't understand why this is true or where this takes me?
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[rude man]

You have an equation and you don't know what the symbols represent? May I suggest your first step is to find that out.
(Hint: the problem tells you what I is; it's given.)

 

[Ben Whelan]

You have an equation and you don't know what the symbols represent? May I suggest your first step is to find that out.
(Hint: the problem tells you what I is; it's given.)

Sorry i should have qualified, I know what the symbols represent, and i realize that I_d is given, i also assume I is 0. However what i don't understand is why?

 

[rude man]

Sorry i should have qualified, I know what the symbols represent, and i realize that I_d is given, i also assume I is 0. However what i don't understand is why?

Actually, I is given, not I_d. I is the current in the wiring, not inside the capacitor. Inside the capacitor, I = 0 as you say.
Your answer is to be B(I,r).
Do you know the Maxwellian extension of Ampere's law? It relates the circulation of B to the electric flux inside the circulation perimeter.
Hint: the perimeter is itself a function of r. It is inside the circular plates.

 

[HalfLostAndSearching]

I can provide a response to the content provided. First, let's start with Maxwell's extension to Ampere's law, which states that the line integral of the magnetic field around a closed loop is equal to the permeability of free space (μ0) times the total current passing through the loop, including any displacement current (Id) which is caused by changing electric fields.

In this scenario, a cylindrical parallel plate capacitor is being discharged by an external current I. As the capacitor is being discharged, the total electric field flux inside the capacitor is changing at a rate of dΦe/dt, which is equal to the external current I divided by the permittivity of free space (ε0). This change in electric field flux will induce a displacement current Id, which is given by the equation Id = ε0 dΦe/dt.

Now, in order to determine the strength of the resulting magnetic field B(r, I) inside the capacitor at a radial distance r from the center axis, we can use the equation B⋅2πr = μ0 (I + Id) from Maxwell's extension to Ampere's law. This equation takes into account both the external current I and the displacement current Id, which are both contributing to the total magnetic field inside the capacitor.

So, to summarize, the strength of the resulting magnetic field inside the capacitor is given by the equation B(r, I) = μ0 (I + Id)/2πr, where Id = ε0 dΦe/dt. This equation takes into account both the external current and the displacement current, which are both important factors in determining the strength of the magnetic field inside the capacitor.