# Magnetic field due to displacement current and Ampere's law

• crick
In summary, the conversation discusses the calculation of the magnetic field at different distances both inside and outside a capacitor with plates at a distance d being charged. Two different equations are used to calculate the magnetic field, one for the outside and one for the inside. It is noted that the magnetic field is different inside and outside the capacitor, and this is due to only considering a part of the displacement current. To ensure continuity of the magnetic field at r=R, the two equations can be set equal, leading to a relation between i and V.
crick
I have a doubt on this calculation of magnetic field in presence of displacement current. Consider a capacitor of radius ##R## and with plates at distance ##d## being charged: there is a displacement current in it.

Suppose that I want to calculate the magnetic field ##B## at a distance ##r < R## both inside and outside the capacitor.

Consider the two amperian-loops, both of radius ##r<R##, one inside and one outside the capcitor.

Outside: Ampere Law leads to
$$2 \pi r B= \mu_0 i_C \to B(r)=\frac{\mu_0 i_c}{2 \pi r} \tag{1}$$

Inside: Ampere Law leads to

$$2 \pi rB= \mu_0 \epsilon_0 \frac{\partial \Phi(E(t))}{\partial t}=\mu_0 \epsilon_0 \pi r^2 E(t)\to B(r)=\frac{\mu_0 \epsilon_0 r}{2} E(t)=\frac{\mu_0 \epsilon_0 r}{2d} V(t)\tag{2}$$

Where ##V(t)## is the voltage difference in the capacitor.

Therefore the two different magnetic fields, inside and outside the capacitor are different. Is that right?

I think it is not so strange since ##r<R## and therefore I did not consider all the displacement current, but only a part of it. Or am I missing something important?

What you have done is correct, except in the second equations your should have dE/dt and dV/dt instead of E and V. But the magnetic field needs to be continuous at r=R. Try setting your two expressions equal at r=R and see what you get. What does this tell you about the relation between i and V?

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## What is the concept of displacement current?

Displacement current is a term used in electromagnetism to describe the flow of electric charge that appears to exist in a region of space where there is a changing electric field. It is not a physical current like conduction current, but rather an electric field that changes over time.

## How is displacement current related to Ampere's law?

Displacement current is a crucial component of Ampere's law, which states that the magnetic field around a closed loop is proportional to the current passing through the loop. In the case of displacement current, this current is not due to the flow of charges, but rather the change in the electric field within the loop.

## What is the difference between displacement current and conduction current?

The main difference between displacement current and conduction current is that displacement current is not a physical flow of electric charges, but rather a change in the electric field. Conduction current, on the other hand, is the actual movement of electric charges through a conductive material.

## How is displacement current measured?

Displacement current is not directly measurable, as it is an effect of changing electric fields. However, it can be indirectly measured by observing the magnetic field created by the changing electric field, which follows the principles of Ampere's law.

## What are some practical applications of the concept of displacement current?

Displacement current has many practical applications, such as in the design of electronic devices, transmission of radio waves, and in medical imaging techniques such as magnetic resonance imaging (MRI). It is also an essential concept in the field of electromagnetism, helping to explain the behavior of electric and magnetic fields in various situations.

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