Mutual inductance arises when a current in one circuit produces a changing magnetic field that induces an emf in another circuit. On the other hand, self-inductance arises when the current passing through the circuit changes, creating a changing magnetic flux, resulting in inductance in the same circuit.
Consider a circuit connected to an AC source. As the current varies with time, the magnetic flux through the circuit correspondingly changes. Faraday's law tells us that an emf would therefore be induced in the circuit, which will be given by a negative time derivative of the magnetic flux. Since the magnetic field due to a current is directly proportional to the current, the flux due to this field is also proportional to the current. The proportionality constant is known as the self-inductance of the circuit. Similar to mutual inductance, the SI unit of self-inductance is henry.
Self-inductance is associated with the magnetic field produced by a current; any configuration of conductors possesses self-inductance. For example, besides a wire loop, a long, straight wire also has self-inductance, as does a coaxial cable. A coaxial cable is most commonly used by the cable television industry and may also be found connecting your cable modem. Coaxial cables are used due to their ability to transmit electrical signals with minimal distortions. Coaxial cables have two long cylindrical conductors that possess current and a self-inductance that may have undesirable effects.
If the current-carrying wire is made into N number of turns, then the self-inductance is expressed as the ratio of N times the magnetic flux through each turn to the current passing through the loop.