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## Monday, December 17, 2007

### AP Physics C: Inductance- Equations to be remembered

The following equations are to be remembered in this section:

1. Self inductance (coefficient of self induction) ‘L’ of a circuit is defined in terms of magnetic flux linkage as

L = Ф/I where Ф is the magnetic flux (in weber) linked with the circuit when a current I ampere flows in the circuit.

L is defined in terms of the self induced voltage ‘ε’ as

L =ε /(dI/dt) where dI/dt is the rate of change of current in the circuit.

The induced voltage opposes the change in the current in the circuit in accordance with Lenz’s law and hence the above equation has to be strictly

L = ε/(dI/dt). But remember that self inductance is a positive quantity. It is the electromagnetic analogue of mass (which has the property of inertia) in mechanics.

[Induced voltage (emf) is represented by the symbol V also, instead of ε].

It will be better to remember the above equation as

ε = –L(dI/dt).

2. An inductance L carrying a current I possesses energy UL given by

UL = ½ LI2, which is stored in the magnetic field established by the current.

3. Self inductance of a infinitely long straight air-cored solenoid is given by

L = μ0n2A where ‘ℓ’ is the length of the solenoid, ‘A’ is its cross section area, ‘n’ is the number of turns per metre and μ0 is the permeability of air (or free space).

[By the term infinitely long solenoid, we mean a solenoid with radius negligibly small compared to the length]

Note that the self inductance is proportional to the square of the number of turns.

4. Mutual inductance (coefficient of mutial induction) is defined in terms of magnetic flux linkage as

M = Ф/I where Ф is the magnetic flux (in weber) linked with the secondary circuit when a current I ampere flows in the primary circuit.

M is defined in terms of the induced voltage ‘ε’ in the secondary circuit as

M = ε /(dI/dt) where dI/dt is the rate of change of current in the primary circuit.

The induced voltage opposes the change in the current in the primary circuit in accordance with Lenz’s law and hence the above equation has to be strictly

M = ε /(dI/dt). But remember that mutual inductance is a positive quantity.

It will be better to remember the above equation as

ε = –M(dI/dt).

5. Mutual inductance between an infinitely long straight solenoid and a short secondary coil wound round it (outside) at the middle is given by

M = μ0nNA where 'n' and A are respectively number of turns per metre and the cross section area of the solenoid (primary) and N is the total number of turns in the secondary coil.

[If the secondary coil is inside the solenoid, the cross section area of the secondary coil is to be used in place of A].

6. Mutual inductance M is related to the primary and secondary self inductances L1 and L2 as

M = K√(L1L2) where K is the coupling coefficient which can have a maximum value of one. This happens when the entire magnetic flux produced by the primary is linked with the secondary.

Note that self inductance and mutual inductance are directly proportional to the permeability of the core material. If a coil is wound on a core of relative permeability μr, its inductance will be μr times the value with air core. Further, the self inductance of a given coil (with a given core) is a constant where as the mutual inductance between two given coils depends on the relative disposition of the coils.

7. Growth of current in a series LR circuit connected in series with a direct voltage is exponential and is given by

I = I0(1– e–Rt/L) where e is the base of natural logarithms and I0 is the final current (at infinite time) given by I0 = V/R, where V is the direct supply voltage.

8. Decay of current in the LR circuit is exponential and is given by

I = I0e–Rt/L

The growth and decay of current are shown in the adjoining figure.

9. Time constant of LR circuit = L/R

[You must remember the time constant of a CR circuit also, which is CR. The growth of charge Q on a capacitor connected in series with a direct voltage source of emf V volt is given by

Q= Q0(1– e–t/RC) where Q0 is the final (maximum) charge equal to CV which is attained at infinite time.

When the capacitor having charge Q0 is allowed to discharge through a resistance R, the decay of charge on the capacitor is given by

Q = Q0e–t/RC

The growth and decay of charge are shown in the adjoining figure.

10. Frequency of oscillations generated when a charged capacitor of capacitance C is discharged through an inductor of inductance L is given by

f = 1/2π√(LC)

The above equation can be written in terms of the angular frequency as

ω = 1/√(LC)

[The above equation is obtained by solving Kirchoff’s voltage equation for the LC circuit:

L(dI/dt) + Q/C =0, which can be rewritten as

L(d2Q/dt2 ) + Q/C =0.

Therefore, d2Q/dt2 = – Q/LC, which is similar to the equation of the simple harmonic motion of a mass ‘m’ attached to a spring of force constant ‘k’, written as

m (d2x/dt2) = – kx

or, d2x/dt2 = –ω2x, where ω = √(k/m) is the angular frequency of mechanical oscillations of the spring-mass system.

The angular frequency of oscillations of the LC circuit (angular frequency of variation of charge on the capacitor) is similarly given by

ω = 1/√(LC)]

In the next post, questions (involving inductance) of the type you can expect in AP Physics C Exam will be discussed.