Magnetic fields due to currents — common mistakes

Always remember to apply the Right-Hand Grip Rule consistently for both straight wires and solenoids (to determine polarity). Double-check the direction of current for parallel wires; 'same direction' means attraction, 'opposite direction' means repulsion – this is a common point for losing marks if confused!

A current-carrying wire produces a magnetic field because electric current is a flow of charged particles (electrons). According to the principles of electromagnetism, any moving charge generates a magnetic field in the space surrounding its path.

Inside a long solenoid, the magnetic fields from each individual turn of the coil add up constructively. The components of the field perpendicular to the axis largely cancel out, while the components parallel to the axis reinforce each other. This results in nearly parallel, equally spaced magnetic field lines, indicating a strong and remarkably uniform magnetic field within its core.

The permeability of free space ($\mu_0$) is a fundamental physical constant that quantifies the ability of a vacuum to support the formation of a magnetic field. It acts as a proportionality constant in equations relating magnetic fields to electric currents, essentially defining the strength of the magnetic field produced by a given current in a vacuum.

The formula is derived by considering one wire creating a magnetic field, and the second wire experiencing a force due to that field. The field from wire 1 is $B_1 = (\mu_0 I_1)/(2\pi r)$. The force on a length L of wire 2 in this field is $F_2 = B_1 I_2 L$. Substituting $B_1$ gives $F_2 = ((\mu_0 I_1)/(2\pi r)) I_2 L$. Rearranging for force per unit length gives $F/L = (\mu_0 I_1 I_2)/(2\pi r)$.