Using method of dimensions, establish relation among given physical quantities :
(a) Heat `H` produced in a wire depends on current `i` flowing through it, its resistance `R` and time `t`.
(b) The magentic field `F_(m)` acting on a moving charged particle in magnetic field depends on charge on particle `q`, its speed `v` and magnetic field `B`
Dimensional formulae : `R : ML^(2) T^(-3) A^(-2), B : MT^(-2) A^(-1)`

Explanation

(a) `H prop i^(a) R^(b) t^(c )`
`[ML^(2) T^(-2)] prop [A]^(a) [ML^(2) T^(-3) A^(-2)]^(b) [T]^(c )`
`M^(1) L^(2) T^(-2) A^(0) prop M^(b) L^(2b) T^(-3b + C) A^(a - 2b)`
Comparign power of `M, T` and `A`
`b = 1`
`-3 b + c = - 2 implies c = 1`
`a - 2b = 0 implies a = 2`
`H prop i^(2) Rt implies H = k i^(2) R t`
From experiments, `k = 1`
`H = i^(2) R t`
(b) `F_(m) prop B^(a) q^(b) v^(c )`
`MLT^(-2) prop [MT^(-2) A^(-1)]^(a) [A^T]^(b) [LT^(-1)]^(c )`
`M^(1) L^(1) T^(-2) A^(0) prop M^(a) L^(c ) T^(-2a + b -c) A^(-a + b)`
Comparing powers `M, L` and `A`
`a = 1`
`c = 1`
`-a + b = 0 implies b = 1`
`F_(m) prop Bqv implies F_(m) = k Bqv`
From experiment, `k = 1`
`F_(m) = Bqv`