SME
University Physics II: Electricity & Magnetism · Problem type
A rod sliding on rails in a magnetic field
Use this for “A 0.6 m rod is pulled at 3 m/s along rails in a 0.8 T field, closed by a 4 Ω resistor. Find the emf, the current and the force needed.”
Diagram
ε = BLv = 0.5 × 0.4 × 5 = 1 V
I
=
ε
R
=
0.5
A
F = BIL = 0.1 N, against v (Lenz)
P = Fv = 0.5 W, the I²R heating in R
This is an example. Type your own values to start a new problem.
ε (
)
=
B (
)
×
L (
)
×
v (m/s)
R
Resistance
example
I
Current
calculated
F
Force needed
calculated
P
Power put in
calculated
Formulas
ε
=
B
×
L
×
v
1 = 0.5 × 0.4 × 5
I
=
ε
R
0.5
=
1
2
F
=
B
×
I
×
L
0.1 = 0.5 × 0.5 × 0.4
P
=
F
×
v
0.5 = 0.1 × 5
P
=
I
2
×
R
0.5
=
0.5
2
×
2
Assumptions
•
B is even and square to the loop; the rod and rails have no resistance, so R is the whole loop’s resistance.
•
The loop’s area grows by Lv each second, so the flux Φ = BLx changes at BLv and Faraday’s law gives ε = BLv; Lenz sets the current’s way round.
•
The field pushes on the current in the rod with F = BIL, against v: a pull of the same size keeps the speed steady.
•
The pull’s power P = Fv all turns to heat in the resistor, so P = I²R too.
Step-by-step
WE KNOW
B = 0.5 T, L = 0.4 m, v = 5 m/s, R = 2 Ω
FIND
induced emf (ε), current (I), force needed (F), power put in (P)
Step 1 · Find induced emf (ε)
Use ε = B × L × v
Faraday’s
law:
ε
=
dΦ/dt
=
d(BLx)/dt
=
BL
×
dx/dt
=
BLv.
ε
=
0.5
×
0.4
×
5
ε
=
0.2
×
5
ε
=
1
V
Step 2 · Find current (I)
Use I = ε ÷ R
Ohm’s
law
round
the
loop:
the
emf
drives
I
through
R
.
I
=
1
2
I
=
0.5
A
Step 3 · Find force needed (F)
Use F = B × I × L
The
field
pushes
on
the
current
in
the
rod
against
v
;
the
pull
must
match
it
to
keep
v
steady.
F
=
0.5
×
0.5
×
0.4
F
=
0.25
×
0.4
F
=
0.1
N
Step 4 · Find power put in (P)
Use P = F × v
The power of the pull is force times speed.
P
=
0.1
×
5
P
=
0.5
W
Check
Put the numbers back in. Both sides should match.
1 = 0.5 × 0.4 × 5 ✓
0.5
=
1
2
✓
0.1 = 0.5 × 0.5 × 0.4 ✓
0.5 = 0.1 × 5 ✓
0.5
=
0.5
2
×
2
✓
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Magnetic fields and induction
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Two parallel wires: the field and the force between them
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