Self-Inductance & Energy

PHYInductance & Transformers

Self-induction is a coil opposing changes in its OWN current by inducing a back-EMF; the proportionality constant is the self-inductance $L$ (henry). Students find the induced EMF from $\varepsilon = -L\,dI/dt$ , compute a solenoid's $L$ from its geometry, and evaluate the magnetic energy $\tfrac12 LI^2$ stored in it.

These are short, reliable numerical marks.

Self-induced (back) EMF

\varepsilon = -L\,\dfrac{dI}{dt}

L

= self-inductance (H),

dI/dt

= rate of change of current (A/s); the minus sign shows the EMF opposes the change (Lenz).

Self-inductance of a solenoid

L = \dfrac{\mu_0 N^2 A}{l}

\mu_0 = 4\pi\times10^{-7}\,\text{T m/A}

N

= total turns,

A

= cross-sectional area (m

^2

l

= length (m); valid for an air-cored solenoid.

Energy stored in an inductor

U = \dfrac{1}{2} L I^2

U

= energy (J) stored in the magnetic field,

I

= current (A); analogous to

\tfrac12 CV^2

for a capacitor.

Self-inductance $L$ depends only on geometry and the core material ( $\mu$ ), not on the current or the EMF.
For the solenoid, $L \propto N^2$ , so doubling the turns quadruples $L$ — a very common conceptual MCQ.
$1\,\text{H} = 1\,\text{V s/A} = 1\,\text{Wb/A}$ ; check units when an answer looks off by a power of ten.
Always convert area from cm $^2$ to m $^2$ ( $1\,\text{cm}^2 = 10^{-4}\,\text{m}^2$ ) and length to metres before substituting.
The stored energy $\tfrac12 LI^2$ resides in the magnetic field; the corresponding energy density is $u = B^2/2\mu_0$ .
Magnitude of the self-induced EMF is $|\varepsilon| = L\,|dI/dt|$ — the sign just indicates opposition; report a positive magnitude unless direction is asked.
Inserting a soft-iron core multiplies $L$ by the relative permeability $\mu_r$ , since $\mu = \mu_r\mu_0$ replaces $\mu_0$ .

Where the marks go

Using $N$ instead of $N^2$ in the solenoid formula for $L$ .
Forgetting to convert $A$ from cm $^2$ to m $^2$ , giving an answer wrong by $10^4$ .
Writing the energy as $LI^2$ or $\tfrac12 LI$ instead of $\tfrac12 LI^2$ .
Confusing $L$ (self-inductance, depends on geometry) with the EMF $\varepsilon$ (depends on $dI/dt$ ).

How the board asks it

Numericalenergy stored in an inductor
A coil of self-inductance $L = 200\,\text{mH}$ carries a steady current of $4\,\text{A}$ . Calculate the magnetic energy stored in the coil.
Numericalself-induced emf from $dI/dt$
The current in a coil of self-inductance $0.5\,\text{H}$ changes uniformly from $2\,\text{A}$ to $8\,\text{A}$ in $0.1\,\text{s}$ . Calculate the magnitude of the self-induced emf.
Numericalself-inductance of a solenoid from geometry
An air-cored solenoid of length $0.5\,\text{m}$ and cross-sectional area $20\,\text{cm}^2$ has $1000$ turns. Calculate its self-inductance $L$ .
Derive / proveenergy $\tfrac12 LI^2$ by integrating instantaneous power
Obtain an expression for the energy stored in an inductor of self-inductance $L$ carrying a steady current $I$ .
Give reasons $L \propto N^2$ for a solenoid
The number of turns of a solenoid is doubled while its length and area are kept unchanged. With reason, state how its self-inductance changes.
Define / statedefinition of self-inductance and the henry
Define the self-inductance of a coil and state its SI unit, the henry, in terms of $\text{Wb}$ and $\text{A}$ .

Practise this topic

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Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.