Self And Mutual Inductance Of Simple Configurations

Mutual inductance between two coils is the property of the system that quantifies how much the magnetic flux produced by one coil links with the second coil. It is defined as the ratio of the induced emf in coil 2 to the rate of change of current in coil 1. The mutual inductance between two coaxial solenoids can be derived as follows: Let:
- be the number of turns of the first solenoid,
- be the number of turns of the second solenoid,
- be the length of each solenoid,
- be the radius of the first solenoid,
- be the radius of the second solenoid,
- be the permeability of the material inside the solenoids. The magnetic field produced by the first solenoid inside it is given by Ampere’s Law: The flux linkage of the second solenoid is: where is the cross-sectional area of the second solenoid, given by . Thus, the flux linkage becomes: The induced emf in the second solenoid is: The mutual inductance is then defined as: Substituting for : This is the expression for the mutual inductance between two coaxial solenoids.

(I) Magnitude of EMF Induced with Rate of Change of Current:

The induced EMF in an inductor is related to the rate of change of current by the formula:

where is the inductance of the inductor.

Since the rate of change of current is the same for both inductors, the induced EMF is directly proportional to the inductance.

Therefore, the inductor with a larger inductance will have a higher EMF. Specifically, if one inductor has a self
-inductance of 20 mH and the other 10 mH, the EMF in the 20 mH inductor will be twice as large.

The graph of EMF vs. time will show this proportional difference.

(II) Energy Stored in Each Inductor with Current Flowing Through It:

The energy stored in an inductor is given by:

where:

• is the energy
• is the inductance
• is the current

For the same current, the energy stored is directly proportional to the inductance. Thus, the inductor with 20 mH stores twice the energy as the 10 mH inductor.

The graph of energy vs. current will show a quadratic relationship, with the curve for the larger inductance always being higher by a factor of 2 for the same current.

Self
-inductance of a coil is a property of the coil that describes its ability to oppose changes in current flowing through it. When a current flows through a coil, it creates a magnetic flux. The self
-inductance is defined as the ratio of the induced emf (electromotive force) in the coil to the rate of change of current. Mathematically, it is given by: where:
- is the number of turns in the coil,
- is the magnetic flux through each turn,
- is the current flowing through the coil. Now, let’s derive the expression for the energy required to build up a current in a coil of self-inductance . Energy required to build up a current: The work required to increase the current by an infinitesimal amount in the coil is given by the product of the induced emf and the infinitesimal current change : From Faraday's law, the induced emf is: Substituting this in the above equation: Since is the rate of change of current, we need to integrate to find the total work required to build up the current from 0 to . The total work done (or energy) to establish the current is: Solving the integral: Thus, the energy required to build up a current in a coil of self-inductance is: This is the expression for the energy stored in the magnetic field of the coil as the current increases.

To determine the number of turns in the solenoid, we can use the formula for the inductance of a solenoid: where (inductance), (permeability of free space), is the number of turns, is the cross-sectional area, and is the length of the solenoid.

First, calculate the cross-sectional area : Substitute the known values into the inductance formula: Simplify to solve for :

Thus, the number of turns in the solenoid is approximately 2866.