Lesson 25 (1) Mutual Induction Imagine two wire loops near-­‐by labeled 1 and 2. Loop 1 carries current !! which is changing in time. The magnetic flux it creates through loop 2 is also changing, and creates an emf in loop2 according to Faraday’s law, which drives a current !! . Conversely, a changing current in loop 2 also causes an emf to appear on loop 1. This phenomenon is called mutual induction. Let !!" be the magnetic flux through loop 2 due to the current in loop 1. Since it is expected to be proportional to !! , the ratio !!"
!!" =
!!
can only depend on the geometrical arrangement between the loops. It is called the mutual inductance between the wire loops. Similarly we can define !!"
!!" =
!!
The unit of mutual inductance is the Henry H: ! = !" ! If the loops have multiple turns, !!" and !!" are to be taken as the sum of the fluxes through individual turns, and are thus proportional to the number of turns !! and !! respectively. On the other hand, since !!" is due to the current from loop 2, it is proportional to !! . Similarly !!" should be proportional to !! . Therefore both !!" and !!" are proportional to the product !! !! . It can be shown that !!" = !!" From the relation between the time rate of change of !! and that of in !!" : !!!"
!!!
= !!"
!"
!"
the induced emf in loop 2 can be found using Faraday’s law: !!!
ℇ! = −!!"
!"
1 A rapid change of current in one circuit can give rise to large emf in another circuit. An application of this idea is the ignition coil in a car used to generate the high voltage needed to fire the spark plugs. Example: An !! -­‐ turn coil where !! = 2 is wrapped around a solenoid of length ℓ = 50!" and cross-­‐sectional area ! = 30!"! , wound with !! = 2500 turns of wires. Find the mutual inductance. Solution: Suppose a current !! runs in the solenoid. Using the infinite length approximation, magnetic field it creates inside the solenoid is ! = !!
!!
ℓ
!! Since the flux through each turn of the !! -­‐turn coil is !", the flux through the coil is !!" = !! !" = !!
!! !!
ℓ
!"! !!" =
!!"
!! !!
2×2500
= !!
! = 4!×10!! ×
×30×10!! = 3.8×10!! ! = 38!" !!
!
0.5
(2) Self Induction The current ! in a single wire loop causes magnetic flux ! through the loop itself. When the current changes, the changing flux induces an emf in the loop. This is called self-­‐induction. Similar to mutual inductance, the self-­‐inductance defined by !
! = !
where again for a wire coil of ! turns, the flux ! contains the factor !. Reasoning in a similar way as the earlier example, the self inductance of a very long solenoid of length ! , cross-­‐sectional area A , turn density n , with a core of permeability µ , is L = µ n 2 A! 2 The induced emf from a changing current is ℇ = −!
!"
!"
The negative sign means the following rule for the direction of the emf, which is also in agreement with Lenz law: If the current is increasing, the emf is opposite to the direction of the current. If the current is reducing, the emf is in the same direction of the current. For these reasons, the induced emf is called a back emf. An inductor is a circuit element that has an inductance concentrated within. It acts like a battery with an emf when the current changes. The diagram below shows a symbol of an inductor. With the direction of current indicated in the symbol, the value and direction of the emf are as shown. Using the battery analogy, the increment of potential on traversing an inductor from left to right is shown in the following diagrams: Note that the value of the current ! can be negative in these diagrams. The arrows on the current simply indicate the positive direction in the circuit. 3 Example: For the values of I and L indicated in the diagram, and if the current is increasing at the rate of 2 ampere per second, we have dI
= 2 A / s dt
dI
The induced emf is L = 4 ! 2 = 8V in the direction of the current. dt
Example: For the values of I and L indicated in the diagram, and if the current is reducing (weakening) at the rate of 2 ampere per second, we dI
also have = 2 A / s dt
dI
The induced emf is L = 4 ! 2 = 8V , same as the direction as the previous example. dt
A clearer representation of this situation is to reverse the arrow of the current as shown. We then see that if the current is weakening, the indicated direction of the emf would try to replenish the current as expected from Lenz law. (3) Build up of a Current When the switch in a circuit connecting a resistor to a battery is closed, a back emf immediately develops due to self inductance that cancels the battery emf, so that the current is zero. The current then begins to rise, and the back emf is 4 gradually reduced. After a long time, the current becomes steady, and the back emf disappears. The behaviors of the current and the back emf as function of time since the closing of the switch are as shown. For a quantitative analysis, consider a circuit composed of a battery of emf ℇ, a resistor of resistance ! , an inductor of inductance L and a switch that can be thrown between A and B as shown. When the switch is connected to A, we choose the direction of current and the sense of traversing the loop as indicated, and write Kirchhoff’s loop equation : !"
ℇ − !" − ! !" = 0 , which can be rewritten in the standard form !"
!
ℇ
!" + ! = ! where we have defined !
! = ! The solution with initial condition I = 0 at t = 0 is !
!=
1 − ! !!
!
!"
ℇ!
The back emf is ℇ!"#$ = ! !" = ! ! ! ! !!
!
!
= !! !! ! Thus, the current eventually reaches the steady state value !! = ℇ !, and the time scale to reach steady state is governed by the time constant ! = L R . This time increases with L because a larger self inductance sets up a larger back emf to oppose the change in current. (4) Decay of Current With steady state current running through R and L , the switch is now thrown to the position B. We then have a loop without the battery. The loop equation leads to 5 !" !
+ = 0 !" !
The solution with initial condition I = I 0 at t = 0 is I = I 0 e!t ! Thus the same time scale governs the decay phase and the rise phase of the current. The change of current is not abrupt when the switch is connected. The sluggishness of the current is the result of inductance. (5) Energy stored in Inductor Multiplying the circuit equation !"
ℇ − !" − ! = 0 !"
by I leads to !"
! 1
ℇ! = ! ! ! + !" = ! ! ! + !! ! !"
!" 2
Integrating and using I = 0 at t = 0 , we find !
ℇ!"#
!
=
! !
! !"#
!
!
!
+ ! !!!
The left side of this equation is the total energy delivered by the battery. The first term on the right side is the energy dissipated in the resistor. To preserve conservation of energy, we regard the second term on the right side as the energy stored in the inductor because of the current it carries. Thus, 1
Energy stored in inductor U = LI 2 2
If the inductor is a very long solenoid of length ! , cross-­‐sectional area A , and turn density n , the inductance is L = µ 0 n 2 A! and the current it carries is related to the magnetic field inside by B
I=
µ0 n
Using these in the formula for the energy of the inductor leads to 6 U
B2
=
A! 2µ 0
We can interpret this to mean that the energy of the inductor is stored in the magnetic field inside, with a density B 2 2µ 0 that can be considered a magnetic energy density. (6) Currents and Voltages at Short and Long Times In a circuit with resistors and inductors, the current and voltage across each circuit element can be obtained in a short time or a long time after a switch is closed without resorting to the solution of differential equations. Here long and short times are in comparison with the time constant L R . The method is to apply Kirchhoff’s loop rule, together with the fact that 1. The current through an inductor cannot experience an abrupt change, and 2. After a long time, the induced emf of an inductor is zero because the current is no longer changing. This is illustrated in the circuit shown. At any time after the switch is closed, the left and right loop equations are ℇ − !! !! − !! !! = 0 !! !! − !
! !! − !!
= 0 !"
Immediately after closing the switch, the current remains equal to zero in the inductor because no abrupt change of current can take place. Therefore !! − !! = 0 It follows that 7 !! = !! =
ℇ
!! + !!
and the voltage across the resistor !! is !! = !! !! =
!!
ℇ !! + !!
while the back emf in the inductor obtained from the second loop equation is ! !! − !!
!!
ℇ!"#$ = !
=
ℇ !"
!! + !!
After a long time, steady state is reached, and the current in the inductor is no longer changing. Therefore ! !! − !!
= 0 !"
The right loop equation now gives !! = 0 The inductor behaves like a short circuit, carrying all the current !! that runs through !! . From the left loop equation, this current is ℇ
!! = !!
Suppose now the switch is opened. We then only have one loop as shown. The loop equation is !"
−!!! − ! = 0 !"
where I is the same current running through the resistor !! and the inductor. Since the current through the inductor cannot change immediately after opening the switch, we have 8 !=
The voltage across the resistor R2 is ℇ
!!
!! = !!! =
!!
ℇ !!
and can be much higher than the emf of the battery ℇ. When a circuit running a steady current is broken, the situation is similar to the example above with very large value of R2 . The large potential difference appearing across the gap where the circuit is broken can cause sparking. The very short L R
time constant means that the magnetic energy stored in the inductor is released rapidly. 9