# Induced magnetic field in a cylinder

• Silviu
In summary, the problem asks for an approximation solution to the current density, not the exact solution.

#### Silviu

Hello! I am confused about the first EM problem on http://web.mit.edu/physics/current/graduate/exams/gen2_F00.pdf (page 4, with the cylinder in a magnetic field). The solution can be found http://web.mit.edu/physics/current/graduate/exams/gen2sol_F00.pdf. In part a the solution is straightforward. However I am a bit confused about part b. They divide the cylinder in solenoids and calculate the field a a given point produced by these solenoids. This makes sense. However, when they plug in the value of ##j(t)## they use the value from part a, and I am not sure why we can do this. In a simple RL circuit you have an ODE that you solve to find the current as a function of time. Here if I understand well, they calculate the induced magnetic field created be ##j## from part a, and use that to calculate the new ##j##, but once you take self inductance into account, ##j## from part a is not there anymore. Don't you need to calculate the magnetic field induced by this new ##j## to find ##j## at a later time, and this way you need an ODE? What am I missing here?

You are right if the problem was asking for the exact analytical solution for the current density then you would need to solve a system of partial differential equations (apply Maxwell's equation's in differential form with the boundary conditions imposed by this problem and then solve them, my guess is that the system of PDE's going to be very hard and we can hope only for numerical solutions). (or another method would be to solve an ODE for each solenoid as you said, we would have infinitely many ODE's and they would be coupled due to mutual induction between the solenoids but that's another story).

BUT the problem doesn't ask for the exact analytical solution, it just asks for the next order (of ##\omega##) correction to the current density j. In other words it asks for an approximation solution that gives the j up to the first two orders of ##\omega##, while from taylor's series we know that j will probably contain more orders, possibly infinite.

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Delta² said:
You are right if the problem was asking for the exact analytical solution for the current density then you would need to solve a system of partial differential equations (apply Maxwell's equation's in differential form with the boundary conditions imposed by this problem and then solve them, my guess is that the system of PDE's going to be very hard and we can hope only for numerical solutions).

But the problem doesn't ask for the exact analytical solution, it just asks for the next order (of ##\omega##) correction to the current density j. In other words it asks for an approximation solution that gives the j up to the first two orders of ##\omega##, while from taylor's series we know that j will probably contain more orders, possibly infinite.
Thanks a lot! I feel stupid now for asking this...

Delta2
Nope don't feel stupid, we all sometimes, just need to read more carefully what the problem asks for :D

## 1. What is an induced magnetic field in a cylinder?

An induced magnetic field in a cylinder refers to the creation of a magnetic field within a cylindrical object, such as a wire or tube, when it is placed in an external magnetic field or when there is a change in the current flowing through it. This phenomenon is known as electromagnetic induction and is governed by Faraday's law of induction.

## 2. How is an induced magnetic field created in a cylinder?

An induced magnetic field is created in a cylinder when there is a change in the magnetic flux passing through it. This change in flux can be caused by moving the cylinder in an external magnetic field, changing the strength of the external field, or by changing the current flowing through the cylinder.

## 3. What factors affect the strength of an induced magnetic field in a cylinder?

The strength of an induced magnetic field in a cylinder depends on several factors, including the strength of the external magnetic field, the rate at which the magnetic flux changes, the material and geometry of the cylinder, and the electrical properties of the material.

## 4. What are some real-world applications of induced magnetic fields in cylinders?

Induced magnetic fields in cylinders have many practical applications, such as in generators, motors, transformers, and induction heating. They are also used in devices like magnetic sensors and magnetic levitation systems.

## 5. How can we control or manipulate an induced magnetic field in a cylinder?

The strength and direction of an induced magnetic field in a cylinder can be controlled or manipulated by changing the external magnetic field, the current flowing through the cylinder, or the material and geometry of the cylinder. This allows for precise control in various electromagnetic applications.