Two parallel wires in a horizontal plane carry currents \(I_{1}\) and \(I_{2}\) to the right. The wires each have length \(L\) and are separated by a distance \(d\). (a) What are the magnitude and direction of the ficld due to wire 1 at the location of wire \(2 ?\) (b) What are the magnitude and direction of the magnetic force on wire 2 due to this field? (c) What are the magnitude and direction of the field due to wire 2 at the location of wire \(1 ?\) (d) What are the magnitude and direction of the magnetic force on wire 1 due to this field? (e) Do parallel currents in the same direction attract or repel? (f) What about parallel currents in opposite directions?

: To find the magnitude of the magnetic field created by wire 1 at the location of wire 2, we use the Biot-Savart law for long straight wires: \(B=\frac{\mu_0I}{2\pi d}\) where \(B\) is the magnetic field, \(\mu_0\) is the permeability of free space (\(4\pi\times10^{-7}\space Tm/A\)), \(I\) is the current carried by the wire, and \(d\) is the distance between the two wires. Since both wires are in the same horizontal plane and the currents are in the same direction, the magnetic field due to wire 1 at the location of wire 2 will be in the vertical direction (perpendicular to both wires), using the right-hand rule.

: To find the magnitude of the magnetic force on wire 2 due to the magnetic field created by wire 1, we use the formula for magnetic force on a current-carrying wire: \(F=ILB\sin\theta\) Since the angle \(\theta\) between the direction of the current and the magnetic field is \(90^\circ\), the formula simplifies to \(F=ILB\) Substituting the expression for \(B\) from part (a), we get \(F=I_2L\left(\frac{\mu_0I_1}{2\pi d}\right)\) The direction of the force can be determined using the right-hand rule: Place your right thumb in the direction of the current in wire 2 (\(I_2\)) and curl your fingers in the direction of the magnetic field due to wire 1; your palm will point in the direction of the force acting on wire 2. In this case, the force on wire 2 is in the horizontal direction towards wire 1, meaning that wire 2 is attracted to wire 1.

: Applying the Biot-Savart law for long straight wires again, the magnitude of the magnetic field created by wire 2 at the location of wire 1 is the same as in part (a) since the distance between the wires is the same: \(B=\frac{\mu_0I}{2\pi d}\) The direction of the magnetic field due to wire 2 at the location of wire 1 is also in the vertical direction, perpendicular to both wires, by the right-hand rule.

: The magnitude of the magnetic force on wire 1 due to the magnetic field created by wire 2 is given by the same formula as in part (b), but with the currents \(I_1\) and \(I_2\) reversed: \(F=I_1L\left(\frac{\mu_0I_2}{2\pi d}\right)\) Using the right-hand rule again, we find that the force on wire 1 is in the horizontal direction towards wire 2, meaning that wire 1 is also attracted to wire 2.

: Since both wires experience an attractive force towards each other when their currents are in the same direction, we can conclude that parallel currents in the same direction attract each other.

: If the currents were in opposite directions, the directions of the magnetic fields they create would be reversed. Applying the right-hand rule in this case, we find that the horizontal forces acting on both wires would be directed away from each other, meaning that parallel currents in opposite directions repel each other.