Self-Inductance and Circuits

• LC circuits

Recall: RC circuit, increasing current

I

(

t

)  

R

 1 

e



t

/   ε /R

I

63% 0 1

τ

2

τ

3

τ

4

τ t

Recall: RC circuit, decreasing current 

I e

o 

t

/ 

I

o

I

0.37 I 0 0τ

τ

2τ 3τ 4τ

t

Question:

What happens if we put a capacitor in the circuit, along with R, or L, or both?

LC circuits

• A capacitor is connected to an inductor in an LC circuit • Assume the capacitor is initially charged and then the switch is closed • Assume no resistance and no energy losses to radiation C + The switch is closed at t =0; Find I (t).

L I

Oscillations in an LC Circuit, 1

• The capacitor is fully charged – The energy U in the circuit is stored in the

electric field of the capacitor

– The energy is equal to Q • The switch is closed 2 max /2C – The current in the circuit is zero – No energy is stored in the inductor

Oscillations in an LC Circuit, 2

• The current is equal to the rate at which the charge changes on the capacitor (dQ/dt) – As the capacitor discharges, the energy stored in the electric field decreases – Since there is now a current, some energy is stored in the magnetic field of the inductor – Energy is transferred from the electric field to the magnetic field

Oscillations in an LC Circuit, 3

• The capacitor becomes fully discharged – It stores no energy – All of the energy is now stored in the

magnetic field of the inductor

– This energy is equal to ½ LI 2 – The current reaches its maximum value • The current now decreases in magnitude, recharging the capacitor with its plates having

opposite their initial polarity

Oscillations in an LC Circuit, 4

• Eventually the capacitor becomes fully charged and the

cycle repeats

• The energy continues to oscillate between the inductor and the capacitor • The total energy stored in the LC circuit remains constant in time and equals:

U



U C



U L



Q

2 2

C

 1

L

2

I

2

LC Circuit Analogy to Spring-Mass System • The kinetic energy (½ mv 2 ) of the spring is analogous to the magnetic energy (½ L I 2 ) stored in the inductor • The potential energy ½kx 2 stored in the spring is analogous to the electric potential energy ½CV max 2 stored in the capacitor • At any point in the cycle, energy is shared between the electric and magnetic fields

LC circuits

The switch is closed at t =0; Find I (t).

Looking at the energy loss in each component of the circuit gives us: C + E L +E C =0 Which can be written as (remember, P=VI):

LI dI dt L dI dt



Q C I



Q C

 0  0

L I

Solution

Charge and Current in an LC Circuit

• The charge on the capacitor oscillates between Q max and –Q max • The current in the inductor oscillates between I max and –I max • Q and I are 90 I is zero, etc.

o out of phase with each other – So when Q is a maximum,

Energy in an LC Circuit – Graphs

• The energy continually oscillates between the energy stored in the electric and magnetic fields • When the total energy is stored in one field, the energy stored in the other field is zero

Example 1: A 1.00-μF capacitor is charged by a 40.0-V power supply. The fully charged capacitor is then discharged through a 10.0-mH inductor. Find the maximum current in the resulting oscillations.

Example 2: An LC circuit consists of a 3.30-H inductor and an 840-pF capacitor, initially carrying a 105-μC charge. The switch is open for t < 0 and then closed at t = 0. Compute the following quantities at t = 2.00 ms: a) the energy stored in the capacitor b) the energy stored in the inductor c) the total energy in the circuit.

Solution

Example 3: The switch in the figure below is connected to point a for a long time. After the switch is thrown to point b, what are a) the frequency of oscillation of the LC circuit, b) the maximum charge that appears on the capacitor, c) the maximum current in the inductor d) the total energy the circuit possesses at t = 3.00 s?

Solution

Notes About Real LC Circuits

• In actual circuits, there is always some

resistance

• Therefore, there is some energy transformed to internal energy • The total energy in the circuit continuously decreases as a result of these processes