Dimensional Formula of Electric Field: [M L T⁻³ I⁻¹]

The dimensional formula of electric field is [M¹ L¹ T⁻³ I⁻¹]. It tells you that electric field strength depends on mass, length, time and electric current, with time to the power minus 3 and current to the power minus 1.

Electric field is the force felt by each coulomb of test charge, E = F/q. Force brings [MLT⁻²] and charge brings [IT], and dividing the two gives the result above. Students often expect a T⁻² like force has; the extra power of time comes from the charge in the denominator. The electric field formula page covers the physics and the numbers.

Quick facts

Dimensional formula
[M¹ L¹ T⁻³ I⁻¹]
Defining formula
$E = \frac{F}{q}$
SI unit
N/C, which is the same as V/m
In base units
kg m s⁻³ A⁻¹
Type
Vector, pointing the way a positive charge is pushed
Same dimensions as
Potential gradient, dV/dx

Step by step: dimensional formula of electric field from E = F/q

Start from the definition: $E = \frac{F}{q}$ .
Force: $[F] = [MLT^{-2}]$ , from F = ma.
Charge is current times time, $q = It$ , so $[q] = [IT]$ .
Divide: $[E] = \frac{[MLT^{-2}]}{[IT]} = [M^{1}L^{1}T^{-3}I^{-1}]$ .

The time powers combine as $-2 - 1 = -3$ . Current appears with power minus 1 because it sits below the line. Check each exponent once more before you box the answer; this is where most marks are lost. The dimensional formula of charge page explains why charge is written as [IT] rather than as a base quantity in SI.

A second route: the field as a potential gradient

In a uniform field between two plates, $E = \frac{V}{d}$ . Voltage is work per charge, $[V] = \frac{[ML^{2}T^{-2}]}{[IT]} = [ML^{2}T^{-3}I^{-1}]$ . Dividing by a distance [L] gives $[MLT^{-3}I^{-1}]$ , exactly the same as before.

Getting the same bracket from two different formulas is a good self check in an exam. It also proves something useful about units: the newton per coulomb and the volt per metre must be the same unit, since both reduce to kg m s⁻³ A⁻¹. Use whichever is handier. Plates problems usually give V/m; point charge problems usually give N/C. Potential itself is worked through in detail under voltage dimensions.

Checking a field equation with dimensions

Example: verify Coulomb's law for a point charge, $E = \frac{q}{4\pi\varepsilon_0 r^{2}}$ .

Permittivity of free space carries [M⁻¹L⁻³T⁴I²]; multiplying by r² gives [M⁻¹L⁻¹T⁴I²] for the whole denominator.

Right side: $\frac{[IT]}{[M^{-1}L^{-1}T^{4}I^{2}]} = [MLT^{-3}I^{-1}]$ .

Ignore the 4π, since plain numbers carry no dimensions. Both sides match.

Example: convert a field of 3.0 × 10⁶ N/C (roughly where dry air starts to spark) into V/cm.

1 N/C = 1 V/m, and 1 m = 100 cm, so 1 V/m = 0.01 V/cm.

$3.0 \times 10^{6}\ \text{V/m} \times 0.01 = 3.0 \times 10^{4}\ \text{V/cm}$ .

Quantities with matching dimensions

Where [M L T⁻³ I⁻¹] turns up
Quantity	Formula	Unit
Electric field intensity	$F/q$	N/C
Potential gradient	$dV/dx$	V/m
Field of a charged sheet	$\sigma / 2\varepsilon_0$	N/C

Compare this with nearby electric quantities so you do not swap them. Electric flux multiplies the field by an area and gains L², giving [M L³ T⁻³ I⁻¹]. Potential multiplies by a length, giving [M L² T⁻³ I⁻¹]. The magnetic field B has no L at all: [M T⁻² I⁻¹].

Common errors on this bracket

Writing [MLT⁻²Q⁻¹] in an SI answer. Some older books treat charge as a base quantity Q. In SI the base is current, so convert Q to IT.
Keeping T⁻² from force. Dividing by charge adds one more power of time below the line.
Dropping I entirely. Any electric quantity in SI needs I somewhere in its bracket.
Confusing E with energy. Same letter, very different dimensions: energy is [ML²T⁻²].

Electric and magnetic brackets sit side by side in the dimensional formulas table, so you can compare E with B, V and Φ in seconds.

Frequently asked questions

What is the dimensional formula of electric field?

It is [M¹ L¹ T⁻³ I⁻¹]. Divide the dimensions of force, [MLT⁻²], by the dimensions of charge, [IT].

What is the dimension formula of electric field intensity in terms of charge Q?

If a book treats charge as a base quantity, the bracket is written [M L T⁻² Q⁻¹]. SI picks current instead, so Q is replaced by IT and the bracket becomes [MLT⁻³I⁻¹]. Both describe one physical idea.

Why are N/C and V/m the same unit?

Both reduce to kg m s⁻³ A⁻¹ when you write them in base units. One volt means one joule for each coulomb, and one joule spread over one metre is one newton, so V/m equals N/C.

Is electric field a vector or a scalar?

Electric field is a vector. It points whichever way a tiny positive probe charge would be pushed. Dimensions describe only its size, not its direction.

Do electric field and magnetic field have the same dimensions?

No. Electric field is [M L T⁻³ I⁻¹] while magnetic field is [M T⁻² I⁻¹]. Their ratio E/B has dimensions [L T⁻¹], a speed, which is why E = cB for light.