RC, RLC circuit and Magnetic field

RC Charge relaxationRLC OscillationHelmholtz coils

RC Circuit

• The charge on the capacitor varies with time– q = C(1 – e-t/RC) = Q(1

– e-t/RC) is the time constant

• = RC

• The current can be found I( ) t RCεt e

R

Discharging Capacitor• At t = = RC, the charge

decreases to 0.368 Qmax– In other words, in one time

constant, the capacitor loses 63.2% of its initial charge

• The current can be found

• Both charge and current decay exponentially at a rate characterized by t = RC

I t RCdq Qt edt RC

Oscillations in an LC Circuit

• 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

Time Functions of an LC Circuit• In an LC circuit, charge c

an be expressed as a function of time– Q = Qmax cos (ωt + φ)– This is for an ideal LC circu

it

• The angular frequency, ω, of the circuit depends on the inductance and the capacitance– It is the natural frequency o

f oscillation of the circuit1ω

LC

RLC Circuit

2

2 0d Q dQ QL Rdt dt C

A circuit containing a resistor, an inductor and a capacitor is called an RLC Circuit.Assume the resistor represents the total resistance of the circuit.

RLC Circuit Solution

• When R is small:– The RLC circuit is analogous to lig

ht damping in a mechanical oscillator

– Q = Qmax e-Rt/2L cos ωdt– ωd is the angular frequency of oscil

lation for the circuit and 1

2 212dRω

LC L

RLC Circuit Compared to Damped Oscillators

• When R is very large, the oscillations damp out very rapidly

• There is a critical value of R above which no oscillations occur

• If R = RC, the circuit is said to be critically damped

• When R > RC, the circuit is said to be overdamped

4 /CR L C

Biot-Savart Law

• Biot and Savart conducted experiments on the force exerted by an electric current on a nearby magnet

• They arrived at a mathematical expression that gives the magnetic field at some point in space due to a current

Biot-Savart Law – Equation

• The magnetic field is dB at some point P

• The length element is ds

• The wire is carrying a steady current of I

24ˆIoμ dd

π r

s rB

B for a Circular Current Loop

• The loop has a radius of R and carries a steady current of I

• Find B at point P

2

03 22 22

xIRB

x R

Helmholtz Coils (two N turns coils)

2

03 22 22

xIRB

x R

If each coil has N turns, the field is just N times larger.

20

1 2 3 2 3 222 2 2

20

3 2 3 22 2 2 2

1 12

1 12 2 2

x xN IRB B B

x R R x R

N IRBx R R x xR

0dBdx

2

2 0d Bdx

At x=R/2 B is uniform in the region midway

between the coils.