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RC, RLC circuit and Magnetic
field
RC Charge relaxation
RLC Oscillation
Helmholtz coils
RC Circuit
• The charge on the
capacitor varies with
time
– q = Ce(1 – e-t/RC) =
Q(1 – e-t/RC)
 t is the time constant
• t = RC
• The current can be
found I( t )  ε e
t RC
R
Discharging Capacitor
• At t = 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
I t  
dq
Q t RC

e
dt
RC
• Both charge and current
decay exponentially at a
rate characterized by t =
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
of oscillation of the circuit
ω 1
LC
RLC Circuit
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.
d 2Q
dQ Q
L 2 R
 0
dt
dt C
RLC Circuit Solution
• When R is small:
– The RLC circuit is analogous to
light damping in a mechanical
oscillator
– Q = Qmax e-Rt/2L cos ωdt
– ωd is the angular frequency of
oscillation for the circuit and
 1 R  
ωd  

 
LC
2
L

 

2
1
2
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 R  4L / C
C
• If R = RC, the circuit is
said to be critically
damped
• When R > RC, the circuit
is said to be overdamped
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
μo I ds  ˆr
dB 
4π r 2
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
Bx 

0IR
2
2
2 32
2 x R

Helmholtz Coils (two N turns coils)
If each coil has N turns, the field is just N times larger.
Bx 

0IR 2
2
2 x R

2 32

N 0IR 2 
1
B  Bx1  Bx2 

2
2
2
 x R



N 0IR 2 
1
B
 2
2
2
 x  R

dB
0
dx

32





3
2
 R  x 2  R 2  

 
1
32



3
2

2R 2  x2  2xR


d2B
0
2
dx
1

At x=R/2
B is uniform in the region midway
between the coils.