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Inductance©2011 by Bryan Pflueger

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Mutual Inductance

If two coils of wire are placed near each other and have a current passing through them, they will each induce an emf on one another because they each generate a magnetic flux through each of the coils centers.

To show the proportionality of the induced magnetic field to the current we take into account the number of turns as well as the mutual inductance of the two coils, denoted M21.

Equation for Mutual Inductance

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Mutual Inductance

The emfs have a negative sign because according to Lenses Law they have to oppose any change to the magnetic field.

Units of Mutual Inductance

The units of mutual inductance is the Henry, H, named in honor of the American Physicist Joseph Henry.

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Self InductanceIn mutual inductance we explained how two separate circuits effect one another, but now we will discuss the case in which the circuit induces an emf in itself by means of self inductance.

If a loop of wire with N number of coils is in a circuit and if the current varies then so does the magnetic flux resulting in an induced emf. Once again the induced emf will oppose any change to the current, so it lessens the chance of any fluctuations in the magnitude of the current.

Equation for Self Inductance

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Self Inductance

We already know how to deal with a resistor, battery, and parallel plate capacitors in circuits, but an inductor generates a non-conservative electric field. In order to use them in a circuit we have to determine a general principle which is analogous to Kirchoff's Loop rule.

For a circuit with a varying emf and an inductor there are two electric fields. The first is a conservative electric field, Ec, which is produced by the battery and the second is a non-conservative electric field, En, produced by a varying magnetic flux within the inductor.

For now we will assume the inductor's coils have negligible resistance and therefore only a relatively small electric field is required to move charges throughout the circuit. Since the electric field in the circuit is nonzero charge will begin to build up on the terminals of the inductors, and the net electric field inside the inductor will be zero,Ec + En = 0.

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Self InductanceIf we apply Faraday's Law to the previous scenario depicted here, we can determine there is a true potential difference across the inductor related to the conservative electric field, even though the inductor produces a non-conservative electric field within its coils.

LVarying

emf

x

y

Since the non-conservative electric field produced by the inductor is nonzero everywhere except for inside the inductor we can change the integration from the entire loop just to the segment containing the inductor.

The sum of the electric fields within the inductor are zero, Ec + En = 0, so we can rewrite the equation as:

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Self Inductance

LVarying

emf

x

y

We can conclude that the inductor has a potential difference across its terminals which are related to the conservative electric field and we see that the inductor does not resist the current, rather it resists any change (di/dt).

The integration of this simply is the potential difference at the points x and y along the circuit.

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1 An inductor with inductance L is placed in series with a battery. The equation for the for the current in the inductor is given by the I=be-2t. Which of the following represents the emf induced in the inductor at t=1?

A -2Lb/e

B Lb/e2

C -2Lb/e2

D -Lb/3e3

E zero

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Magnetic Field EnergyA battery has a potential difference of ε, a resistor of resistance r, and an inductor of self inductance L are all placed in a series circuit. The current through the circuit will initially be zero, but it will eventually reach its max value of ε. This is because the inductor is resisting the change in the current, but slowly the current will grow to its max value as will the potential difference across the inductor.

In between these two points in time we can write several equations which will allow us to calculate the power stored in the inductor after a long time.

positive because the current is increasing

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Magnetic Field EnergyThe equation we just found for the potential energy stored in an inductor can be applied to any other shape, such as a toroidal solenoid whose volume is equal to the circumfrence multiplied by its area.

The value of its self inductance is:

The value of its Potential Energy is given by:

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Magnetic Field EnergyThe magnetic energy density is given as the ratio of the potential energy and the volume of the inductor. It is denote by u.

The magnetic energy density can also be represented in terms of the magnetic field. The magnetic field inside the toroidal solenoid is:

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2 An inductor of Inductance .5H is placed in series with a battery which supplies a steady current of 2A. After a long time what is the energy stored within the inductor?

A 1J

B 2 J

C 3 J

D 4 J

E 5 J

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R-L Circuit

R L

Initially when switch S1 is closed the current flowing through that segment of the circuit is zero, but after a long time the current is at its maximum of ε/R; however, we want to discuss the case in between these two points in time.

Initially the current is zero, so the rate of change of current is:

When the current has reached its maximum value, di/dt=0.

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Using Kirchoff's loop rule we know the net voltage drop in the loop must be equal to zero, therefore:

R-L Circuit

Current for a R-L Circuit in terms of time

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R-L Circuit

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R-L CircuitAfter the circuit has reached it maximum current, what happens if we open S1 and close S2? We know that the inductor wants to prevent any change in the current so initially it will remain at Imax, but eventually it will dissipate and drop down to zero.

Current Decay with Respect to time

R L

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R-L CircuitCurrent Decay

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R-L CircuitTime Constant

The R-L Circuit is dependent on R/L. In one time constant the currents value changes by I(1-1/e), when it is connected to the battery and by 1/e, when it is decaying.

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3 An inductor of unknown inductance and a resistor of 12Ω are placed in series with a battery that supplies a current throughout the circuit. The current in the inductor is given by the equation, I=Io-Ioe-3t. What is the value of the inductance?

A .167H

B .25H

C 4H

D 36H

E Not enough information provided

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L-C CircuitIf we place a completely charged capacitor whose potential difference is V and has an initial charge of Q, Q=CV, in series with an inductor an interesting case occurs. Since the capacitor stores energy in its electric field and the inductor will store the energy in its magnetic field, this will cause an oscillation of the energy back and forth from the inductor to the capacitor.

As the Capacitor discharges the current will slowly reach its maximum value, then it will be stored in the inductor's magnetic field. After the current has reached its maximum value current will continue to flow and start to recharge the capacitor, but with opposite polarity. This will be a continuous process that will constantly reverse the polarity of the capacitor.

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L-C CircuitUsing Kirchoff's Loop Rule we set up an equation for the net voltage drop in the circuit

Divide by -L and sub in for di/dt

This is the same equation you would derive for simple harmonic motion, as well as an equation for its position and its angular frequency.

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L-C CircuitBy determining what is analogous between the components of the Inductor-Capacitor Circuit and the Mass-Spring we can determine an equation for the current.

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L-C CircuitBy confirming that the equations are analogous for the circuit and the mass spring system, we can also setup one for the energy of the system. Remember charge Q is the max amount of charge, and q is just the charge sometime during the discharge and charging states.

Energy in conserved in the circuit, and by solving for the current we can find out how it is dependent on the charge on the capacitor.

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4 A fully charged capacitor and an inductor are placed in series with each other and have a frequency of ω. If the inductance was quadrupled and the capacitance was cut in half, which of the following represents the new frequency?

A

B

C

D

E

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L-R-C CircuitBefore in the L-C Circuit the energy was conserved, but what if a resistor were place in series with the other two components to form an L-R-C circuit. In this case the energy would continue to dissipate as it passes through the resistor each time.

When the value of r still allows for some oscillation of the energy it is considered to be underdamped. If the circuit no longer oscillates then it is considered critically damped. If there is an even greater resistor in the circuit it will be overdamped and the capacitor will approach zero even faster.

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Q

0 t

L-R-C CircuitCharge with Respect to Time