Exploring Ampere's Law in a Radial Current Density Cylinder: Quick Test Prep

In summary, the conversation discusses the application of Ampere's Law on a solid cylindrical wire with a varying radially current density. The formula used is B*2*pi*r= U0* \intJ dA, and the question is asked why dA = 2*pi*r*dr. The group discusses the visualization of dA and whether the current density will vary along the length of the cylinder or always be radial. The purpose of the conversation is to gain a quick understanding of the magnitude of the magnetic field outside the wire, for a test the next day.
  • #1
StephenDoty
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Ampere's Law question (need quick answer test tomorrow)

Using Ampere's Law on a solid cylindrical wire with radius R and a current density in the direction of the symmetry axis of the wire. The current density varies radially. J=J0*r^2. What is the magnitude of the the magnetic field when r>R, outside the wire?

Using the formula B*2*pi*r= U0* [tex]\int[/tex]J dA. Which equals B*2*pi*r = U0 * [tex]\int[/tex] J0* r^2 dA.

Now why does dA = 2*pi*r dr?
Can you guys give me an explanation? How does the information that the current density varies radially show that dA= 2*pi*r*dr? I am having trouble visualizing dA.

If the current density varied along the length of the cylinder wouldn't dA= pi*r^2dx? Would the current density vary along the length of the cylinder or will the current density always vary radially?

Thanks for the help.
Stephen
 
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  • #3
,

Hello Stephen,

Great question! The reason why dA = 2*pi*r*dr is due to the geometry of the cylinder. When we use Ampere's Law, we are essentially calculating the magnetic field at a specific point by integrating the current density over a closed loop around that point. In this case, the loop is a circle with radius r.

Now, when we calculate the area of this circle, we use the formula A = pi*r^2. However, in this case, we are not just looking at the area of the circle, but also the current density J, which varies radially. This means that as we move along the loop, the current density at each point will be different.

To take this into account, we need to multiply the area element dA by the current density at that specific point, which is J0*r^2. This results in dA = J0*r^2*pi*r^2 = J0*pi*r^4.

Now, when we integrate over the entire loop, we get U0 * \int J0*pi*r^4 dr. And since we are integrating over a circle, the limits of integration are from 0 to the radius of the circle, which is r. This results in U0 * J0 * pi * \int r^4 dr = U0 * J0 * pi * (r^5)/5.

Finally, we divide both sides by 2*pi*r to get the magnetic field B, which gives us B = U0 * J0 * r^2 / (2 * 5) = U0 * J0 * r^2 / 10.

I hope this helps to explain why dA = 2*pi*r*dr in this case. And to answer your other question, yes, the current density can vary along the length of the cylinder, but in this case, it is specifically stated that the current density varies radially. This means that as we move along the length of the cylinder, the current density will change, but it will always be in the direction of the symmetry axis of the wire.

I wish you all the best on your test tomorrow!

Best,
 

1. What is Ampere's Law?

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.

2. What is the mathematical formula for Ampere's Law?

The mathematical formula for Ampere's Law is B x 2πr = μ0I, where B is the magnetic field, r is the radius of the loop, μ0 is the permeability of free space, and I is the electric current passing through the loop.

3. How is Ampere's Law used in practical applications?

Ampere's Law is used to calculate the magnetic field around current-carrying conductors, such as wires, coils, and solenoids. It is also used in the design of electromagnets, motors, and generators.

4. What is the difference between Ampere's Law and Faraday's Law?

Ampere's Law deals with the relationship between the magnetic field and electric current, whereas Faraday's Law deals with the relationship between the electric field and changing magnetic field.

5. Can Ampere's Law be applied to all current-carrying conductors?

Ampere's Law can only be applied to steady currents, or currents that do not change over time. It cannot be applied to time-varying currents, in which case Maxwell's equations must be used.

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