Two cylindrical conductors with equal cross sections but different resistivities are put end-to-end. Find the magnitude of the charge that accumulates at the boundary between the conductors if a current I=100A flows in the wires. The wires are made of aluminum and copper with resistivities ρCu=1.7×10−8Ω⋅m and ρAl=2.8×10−8Ω⋅m, respectively. This charge should be very small even for this large current I=100A.

Question

Two cylindrical conductors with equal cross sections but different resistivities are put end-to-end. Find the magnitude of the charge that accumulates at the boundary between the conductors if a current I=100A flows in the wires. The wires are made of aluminum and copper with resistivities ρCu=1.7×10−8Ω⋅m and ρAl=2.8×10−8Ω⋅m, respectively. This charge should be very small even for this large current I=100A.

Answer 1

To find the magnitude of the charge that accumulates at the boundary between the conductors, we need to determine the potential difference across the boundary and then use Ohm's law and the definition of resistance to find the charge.

1. Start by finding the resistances of both wires using the formula:

Resistance = resistivity * length / cross-sectional area

For the copper wire:
ResistanceCu = ρCu * length / cross-sectional area

For the aluminum wire:
ResistanceAl = ρAl * length / cross-sectional area

2. Since the wires are put end-to-end, the total resistance of the combination is the sum of the resistances of each wire:

Total resistance = ResistanceCu + ResistanceAl

3. Next, use Ohm's law to find the potential difference across the boundary:

Potential difference (V) = Current (I) * Total resistance

V = I * (ResistanceCu + ResistanceAl)

4. Finally, calculate the charge accumulated at the boundary using the definition of resistance:

Charge = Potential difference / ResistanceAl

Q = V / ResistanceAl

Now, substitute the given values into the equations:

Given:
I = 100 A (current)
ρCu = 1.7 × 10^-8 Ω⋅m (resistivity of copper)
ρAl = 2.8 × 10^-8 Ω⋅m (resistivity of aluminum)

Note: The length and cross-sectional area of the wires are not given, and we assume the wires have equal dimensions.

1. Calculate the resistances of each wire:
ResistanceCu = ρCu * length / cross-sectional area
ResistanceAl = ρAl * length / cross-sectional area

2. Calculate the total resistance:
Total resistance = ResistanceCu + ResistanceAl

3. Calculate the potential difference:
V = I * (ResistanceCu + ResistanceAl)

4. Calculate the charge:
Q = V / ResistanceAl

By following these steps, you can find the magnitude of the charge that accumulates at the boundary between the conductors.