# Ampere's Law and Magnetic Fields

• datran
In summary, the conversation discusses the use of Ampere's equation to find the value of the outside current density needed to make the magnetic field 0 for r > c in a coaxial cable. It is determined that the net current must equal 0 for H to be 0 and the cross-sectional areas of both the inner and outer conductors are needed to solve the problem. The conclusion is that the currents, not the current densities, cancel each other out to attain H = 0.
datran

## Homework Statement

I have a coaxial cable with current density Jo in the center, with radius a, going in -z_hat direction. This generates a magnetic field. The outside of the cable, radius c, also carries a current density Jout going in the +z_hat direction. This generates its own magnetic field.

Find the value of the outside current density to make the magnetic field 0 for r > c

## Homework Equations

I used Ampere's equation.

## The Attempt at a Solution

I do not use Ampere's equation explicitly (starting from dot product and such), but conclude that H = 0 if the net current = 0. and then find out Jout in terms of Jo.

Is this correct thinking? I mean, if I drew a loop that was outside of the coaxial cable, the Inet would be 0, but there is still an H-field being contributed by both the current densities.

How else would I go solving this problem then?

Thanks!

We need to know the cross-sectional area of the outide conductor.

You are right in assuming the net current = 0 for H to be 0. If the inside conductor cross-sectional area = a then you know the inside current is Joπa2 but we also need to know the outside area & it can't be πc2 obviously.

Ok thank you! This was more of a conceptual question than anything. So if the two current densities are in opposite direction but equal to each other in magnitude, this will cancel out the magnetic field entirely?

I'm lost in the understanding because if we take an amperian loop outside of the coaxial cable and assume that the currents were different (meaning they each generate their own magnetic field, which does not cancel each other out), the enclosed current will be 0, but that does not necessarily guarantee the magnetic field is 0 right?

datran said:
Ok thank you! This was more of a conceptual question than anything. So if the two current densities are in opposite direction but equal to each other in magnitude, this will cancel out the magnetic field entirely?

I'm lost in the understanding because if we take an amperian loop outside of the coaxial cable and assume that the currents were different (meaning they each generate their own magnetic field, which does not cancel each other out), the enclosed current will be 0, but that does not necessarily guarantee the magnetic field is 0 right?

It's not the current densities that cancel each other to attain H = 0, it's the currents. That's why you have to know the cross-sectional area of the outer conductor. Then (area of inner conductor) * Jo = (area of outer conductor) * Jout to give ∫Hds = I = net current = 0 so by symmetry H = 0 everywhere along any closed loop outside the outer conductor.

Your approach is correct. Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the current enclosed by the loop. In this case, since the loop is outside the coaxial cable, the current enclosed is zero. Therefore, the line integral of the magnetic field around the loop must also be zero. This means that the magnetic field outside the cable must be zero.

To find the value of the outside current density, you can use the fact that the magnetic field outside the cable is due to the current density Jout. So, you can set the magnetic field outside the cable equal to zero and solve for Jout in terms of Jo. This will give you the value of Jout that will make the magnetic field outside the cable equal to zero.

Alternatively, you can also use the fact that the total current must be conserved, so the current flowing in the -z_hat direction in the center of the cable must be equal to the current flowing in the +z_hat direction on the outside of the cable. This will also give you the value of Jout in terms of Jo.

Overall, your approach is correct and you have correctly applied Ampere's Law to solve this problem. Good job!

## 1. What is Ampere's Law and how is it related to magnetic fields?

Ampere's Law is a fundamental law in electromagnetism that relates the magnetic field around a closed loop to the electric current passing through that loop. It states that the line integral of the magnetic field around a closed loop is equal to the sum of the enclosed electric currents multiplied by a constant, known as the permeability of free space.

## 2. How is Ampere's Law different from Faraday's Law?

Ampere's Law and Faraday's Law are two different laws that govern the behavior of electromagnetic fields. Ampere's Law deals with the relationship between magnetic fields and electric currents, while Faraday's Law deals with the relationship between changing magnetic fields and induced electric fields. In other words, Ampere's Law explains how electric currents create magnetic fields, while Faraday's Law explains how changing magnetic fields create electric fields.

## 3. What is the significance of Ampere's Law in practical applications?

Ampere's Law is essential in understanding and predicting the behavior of magnetic fields in various practical applications. It is used in the design of electric motors, generators, transformers, and other electromagnetic devices. It is also crucial in the study of magnetism and its effects on materials, such as in the development of magnetic storage devices like hard drives and magnetic resonance imaging (MRI) scanners.

## 4. Can Ampere's Law be used to calculate the magnetic field inside a solenoid?

Yes, Ampere's Law can be used to calculate the magnetic field inside a solenoid, which is a long cylindrical coil of wire that produces a nearly uniform magnetic field when an electric current is passed through it. By considering a closed loop around the solenoid, Ampere's Law can be applied to find the magnetic field inside the solenoid in terms of the number of turns per unit length and the current passing through the solenoid.

## 5. What is the role of Ampere's Law in Maxwell's equations?

Ampere's Law is one of the four Maxwell's equations, which are a set of fundamental equations that describe the behavior of electric and magnetic fields. These equations, along with Ampere's Law, explain how electric and magnetic fields are generated, how they interact with each other, and how they propagate through space. Therefore, Ampere's Law plays a crucial role in understanding and applying Maxwell's equations in various fields, including electromagnetism, optics, and telecommunications.

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