Two wires parallel to each other and separated by a distance d carry different currents; they exert a force with magnitude F on each other.
(a) The wires carry current in opposite directions. If the current carried by both wires is divided by five and the distance between the wires is divided by three, then what will the force's magnitude be (as a multiple of F)?
(b) The wires carry current in the same direction . If the current carried by one of the wires is divided by five and the distance between the wires is divided by five, then what will the force's magnitude be (as a multiple of F)?
To find the answers to these questions, we can use Ampere's Law, which states that the force between two parallel wires carrying currents is directly proportional to the product of the currents and inversely proportional to the square of the distance between the wires.
Let's solve each part step by step:
(a) The wires carry current in opposite directions, and we are asked to divide the current by five and the distance by three. Let's denote the original current as I and the original distance as d.
According to Ampere's Law, the force between the wires is given by:
F = k * (I1 * I2) / d^2
Where k is a constant.
If we divide the current carried by both wires by five, the new currents will be I/5 and -I/5. Similarly, dividing the distance by three, the new distance will be d/3.
So, the new force between the wires can be calculated as:
F_new = k * ((I/5) * (-I/5)) / (d/3)^2
Simplifying the equation:
F_new = k * (I^2/25) / (d^2/9)
Note that we are not concerned with the actual value of the constant k, as we are only interested in the force's magnitude as a multiple of F. Therefore, it cancels out when comparing the new and original forces:
F_new = (I^2/25) / (d^2/9)
The force's magnitude in part (a) will be (I^2/25) / (d^2/9) times F.
(b) The wires carry current in the same direction and we are asked to divide one of the currents by five and the distance by five. Using the same notation as before, the original current is I, and the original distance is d.
According to Ampere's Law, the force between the wires is given by:
F = k * (I1 * I2) / d^2
If we divide one of the currents by five, the new currents will be I and I/5. Dividing the distance by five, the new distance will be d/5.
So, the new force between the wires can be calculated as:
F_new = k * (I * (I/5)) / (d/5)^2
Simplifying the equation:
F_new = k * ((I^2)/5) / (d^2/25)
Again, note that the constant k cancels out when comparing the new and original forces:
F_new = (I^2/5) / (d^2/25)
The force's magnitude in part (b) will be (I^2/5) / (d^2/25) times F.
In summary:
(a) The force's magnitude will be (I^2/25) / (d^2/9) times F.
(b) The force's magnitude will be (I^2/5) / (d^2/25) times F.