Dimensional Formula of Self Inductance: [ML²T⁻²I⁻²]

The dimensional formula of self inductance is [M¹ L² T⁻² I⁻²], also written [ML²T⁻²A⁻²]. Self inductance measures how strongly a coil opposes a change in the current through it, and its SI unit is the henry.

The defining relation is the back emf of a coil, ε = −L ΔI/Δt. Voltage brings [ML²T⁻³I⁻¹], and the rate of change of current brings [IT⁻¹]. Dividing the first by the second gives the result. One warning: the symbol L for inductance clashes with the dimension letter L for length, so keep the two straight when you write brackets. Coils and solenoids are worked through on the self inductance formula guide.

Quick facts

Dimensional formula
[M¹ L² T⁻² I⁻²]
Defining formula
$\varepsilon = -L\frac{\Delta I}{\Delta t}$
Other definition
$L = \frac{N\Phi}{I}$
SI unit
henry (H) = Wb/A = V s/A = Ω s
Base units
kg m² s⁻² A⁻²
Type
Scalar property of a coil

Derivation of the dimensional formula of self inductance

From the back emf.

$\varepsilon = L\frac{\Delta I}{\Delta t}$ , dropping the minus sign, which shows direction only.
So $L = \frac{\varepsilon\,\Delta t}{\Delta I}$ .
Emf is a voltage: [ML²T⁻³I⁻¹]. Time is [T] and current is [I].
$[L] = \frac{[ML^{2}T^{-3}I^{-1}][T]}{[I]} = [ML^{2}T^{-2}I^{-2}]$ .

From flux linkage. $N\Phi = LI$ gives $L = \frac{N\Phi}{I}$ . Magnetic flux is [ML²T⁻²I⁻¹] and N is a count, so dividing by [I] lands on the same bracket. Both routes use brackets explained under voltage and magnetic flux.

The henry in four equal forms

Each defining relation gives the henry a different look. From the emf, 1 H = 1 V s/A. From flux linkage, 1 H = 1 Wb/A. Since one ohm equals one volt for every ampere, 1 H = 1 Ω s. In base units all three are kg m² s⁻² A⁻².

Example: the current in a coil drops evenly from 2.0 A to 0 A over 0.050 s, and a back emf of 8.0 V appears. Find L.

$L = \frac{\varepsilon\,\Delta t}{\Delta I} = \frac{(8.0\ \text{V})(0.050\ \text{s})}{2.0\ \text{A}} = 0.20\ \text{H}$ .

The unit comes out as V s/A, which is the henry, matching the bracket.

Two checks worth practicing for JEE

Example: show that L/R is a time, and find it for the coil above in series with a 10 Ω resistor.

Resistance: [ML²T⁻³I⁻²]. In the ratio the M, L and I powers match top and bottom and cancel, while time goes from −2 over −3, leaving $[T^{-2-(-3)}] = [T]$ .

Value: $\tau = \frac{0.20\ \text{H}}{10\ \Omega} = 0.020\ \text{s}$ . This is the time constant of the LR circuit.

Example: show that $U = \frac{1}{2}LI^{2}$ is an energy, then find U at 2.0 A.

$[ML^{2}T^{-2}I^{-2}][I^{2}] = [ML^{2}T^{-2}]$ , the bracket of energy.

$U = \frac{1}{2}(0.20)(2.0)^{2} = 0.40\ \text{J}$ .

For the resistance bracket in that check, see resistance dimensions.

Inductance beside related quantities

Quantities linked to [M L² T⁻² I⁻²]
Quantity	Relation	Bracket
Self inductance	$N\Phi / I$	[M L² T⁻² I⁻²]
Mutual inductance	$N_2\Phi_2 / I_1$	[M L² T⁻² I⁻²]
Permeability μ₀	Inductance per length	[M L T⁻² I⁻²]
Solenoid inductance	$\mu_0 N^{2} A / l$	[M L² T⁻² I⁻²]

The solenoid row is a nice check. μ₀ has [MLT⁻²I⁻²], N is a count, and area over length adds one L, which gives exactly the inductance bracket. Self and mutual inductance share dimensions because both are flux linkage per current.

Mistakes on inductance brackets

Mixing inductance with length. Both use the letter L; write the word inductance if that helps.
Writing I⁻¹. Voltage already has I⁻¹, and dividing by a rate of current adds another.
Writing T⁻³. That is the voltage power of time; multiplying by Δt raises it to T⁻².
Keeping the minus sign. Lenz's law sets the direction; dimensions ignore it.

Inductance, capacitance and resistance are best revised together, and the dimensional formulas hub lists the trio in one row.

Frequently asked questions

What is the dimensional formula of self inductance?

It is [M¹ L² T⁻² I⁻²]. From ε = L ΔI/Δt, divide the voltage bracket by current per time.

Which SI unit is used for self inductance?

The henry, symbol H. One henry equals one volt second per ampere, one weber per ampere, or one ohm second.

Do self inductance and mutual inductance have the same dimensions?

They match. Each is flux linkage divided by a current, so each carries [ML²T⁻²I⁻²] and both are measured in henry.

Why is L/R measured in seconds?

Dividing the bracket of inductance by the bracket of resistance cancels everything except one power of time. That ratio is the time constant of an LR circuit.

Is self inductance a scalar?

Yes. It is a property of a coil, set by its turns, size and core, with no direction of its own.