Using method of dimensions, establish the relation among the given quantities.
(a) The potential difference `V` across a conductor depends on current `i` flowing in it and resistance of conductor `R`.
(b) The speed of light `c` can be expressed in terms of free space `mu_(0)`.
The energy `U` stored in an inductor is function of inductance `L` and current `i` flowing through it.
(d) The time constant `tau` to `R - C` circuit can be expressed in terms of resistance `R` and capacitance. `C`.
Dimensional formulae:
`V : ML^(2) T^(-3) A^(-1). R: ML^(2) T^(-3) A^(-2)`,
`in_(0) : M^(-1) L^(-3) T^(4) A^(2), mu_(0) : MLT^(-2) A^(2)`,
`L : ML^(2) T^(-2) A^(-2), C : M^(-1) L^(-2) T^(4) A^(2)`

Explanation

(a) `V prop i^(a) R^(b)`
`[ML^(2) T^(-3) A^(-1)] prop [A]^(a) [ML^(2) T^(-3) A^(-2)]^(b)`
`M^(1) L^(2) A^(-1) prop M^(b) L^(2b) T^(-3b) A^(a - 2b)`
Comparing powers of `M` and `A`
`b = 1`
`a - 2b = - a implies a = 1`
`V = iR`
`c prop in_(0)^(a) mu_(0)^(b)`
`[LT^(-1)] prop [m^(-1) L^(-3) T^(4) A^(2)]^(a) [MLT^(-2) A^(-2)]^(b)`
`M^(0) L^(1) T^(-1) A^(0) prop M^(-a + b) L^(-3a + b) T^(4a - 2b) A^(2a - ab)`
Comparing power of `M` and `L`
`- a + B = 0`
`-3 a + b = 1`
`a = - (1)/(2), b = - (1)/(2)`
`c prop in_(0)^(- 1//2) mu_(0)^(-1//2) implies c prop (1)/(sqrt(mu_(0) in_(0))) implies c = (k)/(sqrt(mu_(0) in_(0)))`
From experiments, `k = 1`
`c = (1)/(sqrt(mu_(0) in_(0)))`
(c ) `U prop L^(a) i^(b)`
`[ML^(2) T^(2)] prop [ML^(-2) T^(-2) A^(-2)]^(a) [A]^(b)`
`M^(1) L^(2) T^(-2) A^(0) prop M^(a) L^(2a) T^(-2a) A^(-2 a + b)`
Comparing power of `M` and `A`
`a = 1`
`- 2 a + b = 0 implies b = 2`
`U prop Li^(2) implies U = k Li^(2)`