Explain why two wires, each with a current flowing in it, exert forces on each other even when they are not touching each other.

Each wire carrying a current produces a magnetic field around it. The direction of the magnetic field can be determined using the right-hand rule. When you point your thumb in the direction of the current, the fingers naturally curl around the wire in the direction of the magnetic field lines. The strength of the magnetic field depends on the amount of current passing through the wire and the distance from the wire.

When two wires are near each other, the magnetic field produced by one wire affects the other wire. This interaction leads to a force exerted between the two wires, even if they are not touching each other. Depending on the direction of the currents in the two wires, this force can be attracticve or repulsive.

The Biot-Savart Law states that the magnetic field dB at a point P due to a small segment length (dl) of a current carrying wire is \begin{align*} \textrm{dB} = \frac{\mu_{0}}{4\pi} \frac{I (\textrm{d}\textrm{l} \times \textrm{r})}{r^{3}}, \end{align*} where dB is the magnetic field vector at point P, \(\mu_0\) is the permeability of free space, I is the current magnitude, dl is the length of the wire segment, r is the vector from the wire segment to the point P, and \(\times\) denotes the cross product.

Ampere's Law states that the magnetic field B around a closed loop is proportional to the current I flowing through the loop. Mathematically, \begin{align*} \oint \textrm{B} \cdot \textrm{d}\textrm{l} = \mu_{0} imathcal{I}, \end{align*} where B is the magnetic field, dl is the differential length around the closed loop, and \(\oint\) denotes the integral around the closed loop.

The magnetic force F between two parallel current-carrying wires separated by a distance d can be calculated using the following equation: \begin{align*} \textrm{F} = \frac{\mu_{0}I_{1}I_{2}L}{2\pi d}, \end{align*} where F is the magnetic force, \(\mu_0\) is the permeability of free space, \(I_1\) and \(I_2\) are the magnitudes of the currents in the two wires, L is the length of the wires, and d is the distance between the wires. This equation shows that a force between the wires exists as long as there are currents flowing in both wires. The force is attractive when currents are in the same direction and repulsive when they are opposite. This is why two wires with current flowing in them exert forces on each other even when they are not touching each other.