In this lecture the following are introduced:
• The Mutual Inductance of one inductor wound over another.
• The sign convention for potential difference across a Mutual Inductor.
• The Energy stored in the magnetic field of an Inductor
• The Energy Density of a magnetic field
• Inductive-Capacitative oscillations

The strength of a magnetic field is called its magnetic induction, and is measured in Tesla. Magnetic flux, Φ, is the amount of magnetic induction, Bp passing at right angles through the cross-sectional area of a closed conducting loop, as symbolised in the equations. Magnetic flux has the unit Tesla.m2.

The Mutual Inductance of one inductor wound over another.

An inductor is a coil of insulated conducting wire, as shown in the diagram, often wound around a hollow or solid cylindrical former. When a secondary coil of Nsturns is brought close to a primary coil of Np turns carrying a current ip, the magnetic field of the primary coil will pass through the turns of the secondary coil, as shown.

The current in the primary will produce the same magnetic flux, Φ_p through each of its own turns.
This produces a total magnetic flux of N_pΦ_p through the primary coil.

When the current through the primary coil is changing, the total magnetic flux through the primary coil will be changing and an emf will be induced across the primary coil, given by Faraday's law:

If the two coils are tightly wound over each other so that they essentially have the same cross-sectional area and magnetic induction through them, then the magnetic fluxes through each turn of each coil can be considered to be the same, i.e. .

Any change in primary flux will be echoed in the secondary flux and induce an emf across the secondary coil. This emf will depend on the total secondary flux and be linked to the changing current in the primary coil.

Here, M_sp is called the coeficient of mutual inductance, and has the SI unit of the Henry.

If the coils have their roles interchanged so that there is a changing current in the secondary coil then an emf will appear across the primary coil:

It can be proved by energy considerations that M_sp = M_sp = M.

Example Two circular flat coils are co-axial as shown in the diagram. The smaller (secondary) coil has 25 turns and diameter 20 mm. The larger (primary) coil has 200 turns and diameter 40 mm. Find the mutual inductance of the two coils, assuming the magnetic field of the primary coil is uniform through the secondary coil.

Substituting:

Ideal Mutual Inductance and Self Inductance

In the absence of magnetic materials the Mutual Inductance can (in principle) be calculated from the geometry of the linked coils. However in practice this is usually difficult and it is measured experimentally. In the ideal case

Now from the definition of self inductance: and from the definition of mutual inductance:

eliminating Φ and ip from the two equations on the left: . By a similar argument,

Eliminating the turns ratio, N_s/N_p gives:     or  

In other words, the mutual inductance is the geometric mean of the self inductances.

Example
An ideal mutual inductor is made from a primary coil of inductance 5m0 and a secondary coil of inductance 10m0.
Find the value of the Mutual Inductance.

Sign Convention for Mutual Inductance

A mutual inductor has two coils tightly wound over each other. The diagram has separated them for ease of description. Place a dot on any end of the primary coil with an instantaneous current drawn flowing into the dot. If the winding sense of the coils is known (clockwise or anticlockwise), place a dot on the end of the secondary so that inward secondary current will produce a magnetic field in the same direction as the magnetic field in the primary. If the winding sense is unknown, arbitrarily mark one end of the secondary with a dot. The potential difference across a coil is then: Potential of dotted end - Potential of plain end.

The Energy stored in the Magnetic Field of an Inductor

For the circuit shown, after the switch is moved from 1 to 2:
The instantaneous power received by the resistor, (R·i2) is dissipated as heat. The instantaneous power received by the inductor is not dissipated as heat, but stored in a magnetic field in its interior, and the energy can be recovered.

The instantaneous power stored is

Integrating to find the energy stored: This says that the amount of energy stored in the magnetic field depends on the square of the current passing through it.

If the current increases then extra energy is stored in the magnetic field.

In the graph the potential (dotted line) jumps abruptly but the current (curved line) responds more slowly. The green area shows the difference between an abrupt current change and what actually happens. The green area shows energy being stored in the magnetic field from the current's energy.

If the current decreases then magnetic field energy can be recovered as current.

In the graph the potential (dotted line) drops abruptly but the current (curved line) responds more slowly. The green area shows the difference between an abrupt current change and what actually happens. The green area shows energy being released from the magnetic field in the form of current.

Example
A 20V emf is applied to a series combination of a 5k0 resistor and a 4m0 inductor. Find
(a) the equilibrium value of the energy stored in the magnetic field, and
(b) how many time constants pass before the energy stored reaches its one third of its equilibrium value.

Find the value of the current at one third of the energy stored, and then the equation for the current at any time can be used to find the time involved.

It takes 0.86 time constants to reach one third of the final energy stored.

The Energy Density of a Magnetic Field

Take a long narrow, tightly wound coil with: • a steady current, I • N turns in its • length l, and • cross-sectional area, A. From this, n = N/l, is the number of turns per unit length.

The self inductance of the coil is (from previous lecture) ,

The magnetic induction inside is (from another lecture)
This is (ideally) constant inside the volume of the coil and zero outside it.

This enables an energy density or energy per unit volume to be calculated.
The energy stored per volume of coil is

Using the magnetic field above: J.m-3

Even though this was derived for a specific ideal case it applies to any magnetic field.

Example
A solenoid with 1000 turns has a length of 300 mm and a diameter of 10 mm. A current of 0.5 A flows through it and the magnetic field is assumed uniform inside the solenoid.
Find the magnetic energy density inside the solenoid.

Comparison of Electric and Magnetic Fields

Quantity	Capacitor	Inductor
Potential Difference
Time Constant
Energy Stored
Energy Density

LC Oscillations

In RC transient circuits the capacitor is suddenly charged or discharged. This causes the electric field between the plates to increase or decrease and so store or release energy.

In RL transient circuits the inductor has the current through it suddenly changed. This causes the magnetic field from the current to increase or decrease and so store or release energy.

In LC oscillatory circuits starting at (a) in the diagram below, the release of energy from the electric field of the fully charged capacitor starts a current which stores energy in the magnetic field of the inductor.

When the capacitor has lost all its electrical energy, then the inductor starts to replenish it by releasing the energy it stored in its magnetic field to send a current to charge the capacitor, and so oscillations continue.

This interchange of electric field energy and magnetic field energy is like the interchange of kinetic and potential energy in a pendulum, or a vibrating spring and mass system.

Taking potentials around a loop, with the capacitor fully charged at time zero.

Here ω₀ is the angular frequency in rad.s^-1.

The current in the circuit is:

Comparing the charge stored on the capacitor and the current in the circuit.

The current is a quarter cycle out of phase with the charge stored. When the current is returns to zero the charge has reversed sign on the capacitor.

Energy Storage in the circuit

The energy stored in the electric field of the capacitor is:

The energy stored in the magnetic field of the inductor is:

The total energy stored is:

At any time, the total energy is constant:

Example
A 5m0 inductor and a 20m0 capacitor form an LC circuit. Find
(a) the angular frequency, and
(b) the value of the current, as a fraction of the maximum current, when one quarter of the total energy is in the electric field, and
(c) the time for the current to change from zero to the value in part (b).

Summarising:

The Mutual Inductance of two coils is

In the ideal case, the mutual inductance is the geometric mean of the self inductances i.e.

The potential difference across a coil is: V = V_{dotted end} - V_{plain end}.

The energy stored in the magnetic field of an inductor is Joule

The energy density of a magnetic field is J.m^-3

In LC Oscillations:

and

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