What is the dimensional formula of R in the relation V equals IR?

Answer 1

This is not entirely straightforward. Voltage and Resistance are both forces and thus have the dimensions:-

Mass x Length / Time squared i.e. MLT -2

Electric current (in Amperes) is the number of electric charges (in Coulombs) passing a point in a unit time and so ought to have the dimensions:-

1 (unit) / Time i.e. T -1

Writing out the equation V = IR using these dimensions gives:-

MLT -2 = MLT -3

which is unbalanced. The missing part is the time taken for the current to flow - there is no instantaneous current as such and time is required. While using the equation V = IR gives results that are numerically correct, the dimensional equation only makes sense when the current is considered as dimensionless, so that a current is always "charges per unit time multiplied by time taken" and thus having the dimensions:-

1 (unit) x Time / Time i.e. dimensionless.

This gives a balanced dimensional equation for V = IR as:-

MLT -2 = MLT -2

Answer

You need to go back to the definitions of the three quantities involved.

The ampere is the SI base unit for current, which is defined in terms of the force between parallel conductors (note that an ampere is NOT defined in terms of a coulomb and a second!): "An ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible cross-sectional area and placed one metre apart in a vacuum, would produce between them a force of 2 x 10 (to the power of 7) newtons per unit length'. So the ampere is defined in terms of the newton and the metre (newtons per metre).

The coulomb is the SI derived unit for electric charge, and is defined as 'the charge transported through any cross-section of a conductor in one second by a current of one ampere'. So the coulomb is defined in terms of the second and the ampere (ampere times second).

The newton is the SI derived unit for forces, and is defined as 'the force which, when applied to a mass of one kilogram, will give it an acceleration of one metre per second per second'. So the newton is defined in terms of the kilogram and the second (kilogram per second per second).

The volt is an SI derived unit for potential difference, and is defined as 'the potential difference between two points such that the energy used in conveying a charge of one coulomb from one point to the other is one joule'. So the volt is defined in terms of the joule and the coulomb (joules per coulomb).

The joule is an SI derived unit for work, and is defined as 'the work done when the point of application of a force of one newton is displaced by a distance of one metre in the direction of that force'. So the joule is defined in terms of the newton and the metre (newtons x metre).

The ohm is an SI derived unit for electrical resistance, and is defined as 'the electrical resistance between two points of a conductor, such that when a constant potential difference of one volt is applied between those points, a current of one ampere results'. So the ohm is defined in terms of the volt and the ampere (volts per ampere).

So the breakdown of the ohm into its SI base units will be as follows:

ohm = volt/ampere =[joule per coulomb]/[newton per metre]

All that's left, now, is to replace each of the derived units (joules, coulombs. and newtons), in the above equation, by the corresponding base units (kilograms, metres, seconds), to determine the ohm in terms of SI base units.