Two infinite conductors are placed parallel to each other, 2 cm apart and presenting each one of them a current of 2 A. A) Draw the problem with the corresponding forces and magnetic fields. B) Calculate the force magnitude experienced by these two conductors.
The correct answer and explanation is:
A) Diagram of the Problem
- Two parallel conductors are placed 2 cm apart.
- Each conductor carries a current of 2 A.
- The magnetic field generated by one conductor exerts a force on the other conductor due to the interaction of their magnetic fields.
The conductors will create a magnetic field around them due to the current flowing through them. The magnetic field from one conductor interacts with the current in the second conductor, resulting in a force.
- The magnetic field BB around each conductor is generated by Ampère’s Law and points in concentric circles around the wire.
- If the currents are in the same direction, the magnetic fields will interact in such a way that the two wires will experience an attractive force.
- If the currents are in opposite directions, the force between the wires will be repulsive.
B) Calculation of the Force Magnitude
To calculate the force between two parallel conductors, we use Ampère’s force law. The formula for the force per unit length between two parallel conductors is: F=μ0I1I22πdF = \frac{{\mu_0 I_1 I_2}}{{2 \pi d}}
Where:
- FF is the force per unit length,
- μ0=4π×10−7 T\cdotpm/A\mu_0 = 4 \pi \times 10^{-7} \, \text{T·m/A} is the permeability of free space,
- I1I_1 and I2I_2 are the currents in the two conductors (both 2 A),
- d=2 cm=0.02 md = 2 \, \text{cm} = 0.02 \, \text{m} is the distance between the conductors.
Substituting the given values: F=(4π×10−7)×(2)×(2)2π×0.02F = \frac{{(4 \pi \times 10^{-7}) \times (2) \times (2)}}{{2 \pi \times 0.02}}
Simplifying: F=16π×10−72π×0.02=8×10−70.02=4×10−5 N/mF = \frac{{16 \pi \times 10^{-7}}}{{2 \pi \times 0.02}} = \frac{{8 \times 10^{-7}}}{{0.02}} = 4 \times 10^{-5} \, \text{N/m}
Thus, the force per unit length between the two conductors is 4×10−5 N/m4 \times 10^{-5} \, \text{N/m}.
Conclusion
The force experienced by each conductor due to the magnetic field of the other is 4×10−5 N/m4 \times 10^{-5} \, \text{N/m}. If the currents flow in the same direction, the force will be attractive; if they flow in opposite directions, the force will be repulsive.