RC, RLC circuit and Magnetic field
RC Charge relaxation
RLC Oscillation
Helmholtz 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 RCε
t eR
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 e
dt 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 can be expressed as a function of time– Q = Qmax cos (ωt + φ)
– This is for an ideal LC circuit
• 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
20
d Q dQ QL R
dt 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 oscillation for the circuit and
12 2
1
2d
Rω
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
24
ˆIoμ 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
xIR
Bx R
Helmholtz Coils (two N turns coils)
2
03 22 22
xIR
Bx 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 IR
B B Bx R R x R
N IRB
x R R x xR
0dBdx
2
2 0d B
dx At x=R/2 B is uniform in the region midway
between the coils.