Magnetic field from displacement current in charging capacitor

[PhysMajor34]

Alright, my teacher gave us a review sheet of problems for the test tomorrow, supposedly all the concepts will be covered on the test, and I've got NO CLUE where to even start on this problem. It supposedly has something to do with section 30.7 in Serway/Beichner's physics text, titled "Displacement Current And The General Form Of Ampere's Law", but I can't find anything in there or elsewhere that helps me understand this.
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A .4A current is charging a capacitor that has circular plates 13cm in radius.
(a) If the plate separation is 4.00 mm, what is the time rate of increase of electric field between the plates?
(b) What is the magnitude of the magnetic field between the plates 5.00 cm from the center?
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I can't find any equations relating capicitor charging with magnetic fields, or vice versa... help? Please? I'm on my knees here!

 

[Doc Al]

It supposedly has something to do with section 30.7 in Serway/Beichner's physics text, titled "Displacement Current And The General Form Of Ampere's Law", but I can't find anything in there or elsewhere that helps me understand this.

I don't have that text, but I find that hard to believe, given the title of the section!

I can't find any equations relating capicitor charging with magnetic fields, or vice versa... help?

In a nutshell, the idea is that the changing electric field within the capacitor creates the magnetic field.

You need to apply Ampere's law to find the magnetic field inside the capacitor. But first you have to understand Maxwell's extension of Ampere's law to include the so-called displacement current. This displacement current is proportional to the rate of change of the electric flux. The book should have this, but if not, here's a start:
http://academic.mu.edu/phys/matthysd/web004/L0225.htm

 

[M98Ranger]

Hi there,

I can understand how frustrating it can be to come across a problem that you're not sure how to approach. Don't worry, you're not alone! Let's break down the problem and see if we can figure out a way to solve it.

First, let's focus on the concept of displacement current. This is a term used to describe the flow of electric current that is not carried by the movement of electrons, but rather by the changing electric field. In other words, it is the flow of electric current that is responsible for the creation of a magnetic field.

Now, let's look at the given information. We have a capacitor with circular plates and a current of 0.4A charging it. The first part of the problem is asking for the time rate of increase of electric field between the plates. To solve this, we can use the equation for displacement current:

I = ε₀(dϕE/dt)

Where I is the displacement current, ε₀ is the permittivity of free space, and dϕE/dt is the rate of change of electric flux. We know the current (I) and the plate separation (d), so we just need to find the rate of change of electric flux. This can be calculated by taking the derivative of the electric field with respect to time.

For the second part of the problem, we need to find the magnetic field between the plates at a distance of 5.00 cm from the center. To do this, we can use Ampere's Law, which states that the line integral of the magnetic field around a closed loop is equal to the total current passing through that loop. In this case, the closed loop is a circle with a radius of 5.00 cm, centered at the midpoint between the plates. We already know the current, so we just need to find the line integral of the magnetic field.

I hope this helps you understand the problem a bit better. Remember to always start by identifying the relevant concepts and equations, and then plug in the given information to solve for the unknown variables. Good luck on your test tomorrow!