RC circuits and Magnetism
Dr Jacob Dunningham
School of Physics and Astronomy University of Leeds
EM-L7-1
Review of Lecture 6
• Resistors in series or in parallel
R=
X
X 1
1
=
R
i Ri
Ri
i
• Power dissipated
P =VI
• Kirchhoff’s rules
Reading: Tipler, sections 25-3, 25-4, 25-5
Review
EM-L7-2
Overview of Lecture 7
The plan for todays lecture
• RC circuits
• Introduction to magnetism
• Magnetic force
- point particle
- current element
• Summary
Preparation: Tipler, sections 25-6, 26-1, 26-2
Overview
EM-L7-3
RC circuits
EM-L7-4
Discharging a capacitor
Apply Kirchhoff’s loop rule
Q
−I R = 0
C
Q
dQ
+R·
= 0
C
dt
Differential equation. Need to find the solution Q(t).
RC circuits
EM-L7-5
Discharging a capacitor
Finding the solution Q(t)
dQ
Q
+R·
= 0
C
dt
dQ
1
= −
dt
Q
RC
RC circuits
EM-L7-6
Discharging a capacitor
Finding the solution Q(t)
dQ
Q
+R·
= 0
C
dt
dQ
1
= −
dt
Q
RC
Z Q(t)
Z t
dQ0
1
0
dt
=
−
RC 0
Q(0) Q0
1
t
ln (Q(t)/Q(0)) = −
RC
1 ·t
− RC
Q(t) = Q(0) · e
RC circuits
EM-L7-7
Discharging a capacitor
Finding the solution Q(t)
dQ
Q
+R·
= 0
C
dt
dQ
1
= −
dt
Q
RC
Z Q(t)
Z t
dQ0
1
0
dt
=
−
RC 0
Q(0) Q0
1
t
ln (Q(t)/Q(0)) = −
RC
1 ·t
− RC
Q(t) = Q(0) · e
Time constant τ = R C
− τt
Q(t) = Q(0) · e
RC circuits
EM-L7-8
Discharging a capacitor
The current I(t) is
dQ
Q(0) − t
I(t) = −
=
·e τ
dt
RC
RC circuits
EM-L7-9
Charging a capacitor
Kirchhoff’s loop rule
E − I R − Q/C = 0
dQ
Q
+
dt
C
Solution Q(t), using time constant τ = RC, is
E=R
− τt
Q(t) = C E (1 − e
RC circuits
)
EM-L7-10
Charging a capacitor
Charge in the capacitor Q(t)
− τt
Q(t) = C E · (1 − e
)
Current I(t) = dQ/dt is
E
− τt
I(t) =
· e
R
RC circuits
EM-L7-11
Magnetic Force
EM-L7-12
Magnetic force
~ the
When a charge q moves with velocity ~
v in a magnetic field B
~ is
magnetic force F
~ = q·~
~
F
v×B
~ . The ”right-hand-rule”
The force is perpendicular to ~
v and B
gives the direction of the force.
Magnetic Force
EM-L7-13
Unit
The SI unit of the magnetic field strength is the Tesla (T).
1T=1
N/C
N
=1
m/s
A·m
Examples
Earth magnetic field strength ∼ 100 µT
Strongest electro-magnets ∼ 10 T
Another commonly used unit (cgs-system) of the magnetic field
strength is the Gauss (G)
1 G = 10−4 T
Magnetic Force
EM-L7-14
Magnetic field lines
• The magnetic field lines are always perpendicular to the force
on a moving charge.
• Magnetic field lines form closed loops. There appear to be
no magnetic charges (monopoles).
Magnetic Force
EM-L7-15
Force on a current element
Total force on a wire segment is
~ = (q · ~
~ ) · (n · A · dl)
dF
v×B
Magnetic Force
EM-L7-16
Force on a current element
Total force on a wire segment is
~ = (q · ~
~ ) · (n · A · dl)
dF
v×B
This can be written as
~ = (n · q · v · A) · (d~l × B
~)
dF
Magnetic Force
since:
~
v = v · d~l/dl
EM-L7-17
Force on a current element
Total force on a wire segment is
~ = (q · ~
~ ) · (n · A · dl)
dF
v×B
This can be written as
~ = (n · q · v · A) · (d~l × B
~)
dF
since:
~
v = v · d~l/dl
The current in the wire is
I =n·q·v·A
⇒
~ = I · d~l × B
~
dF
where d~l is the wire segment.
Magnetic Force
EM-L7-18
Motion in a magnetic field
EM-L7-19
Motion in a magnetic field
• The magnetic force is always perpendicular to the velocity
of the particle
• The speed does not change. The kinetic energy is constant.
• In a uniform magnetic field a charged particle follows a circular path.
Motion in a magnetic field
EM-L7-20
Cyclotron motion
~ = m ~a
F
2
v
~ = m·
q~
v×B
r̂
r
v2
qvB = m
r
mv
r =
qB
Motion in a magnetic field
EM-L7-21
Summary
• Charging and discharging a capacitor
τ =R·C
• Magnetic force
~ = q·~
~
F
v×B
• SI unit Tesla
1T=1
N
N/C
=1
m/s
A·m
• Cyclotron motion in a uniform field
mv
r=
qB
Preparation: Tipler, sections 26-2, 27-1, 27-2
Summary
EM-L7-22