Question

1. Hooke's law states that the force F in a spring extended by a length x is given by F= -kx . Calculate the dimension of spring constant k.

2. The force between two wires 1,2 length 1 meters separated by a distance d meters and carrying currents I1 AND I2 Amperes is given by :

`F = (kI_1I_2)/(d)` .
find (a.) the units of constant k
(b.) dimension of k

Answer 1

1. With N = Newtons, m = meters, we have

`F = -kx \iff \mathrm N = -k (\mathrm m) \implies k = (\rm N)/(\rm m)`

N itself can be broken down according to Newton's second law,

`F=ma \iff \mathrm N = \mathrm{kg} \, (\rm m)/(\rm s^2)`

so the dimension of the force
`F` is

`[F] = [M\cdot L\cdot T^(-2)]`

(where M = mass, L = length, T = time) and hence the dimension of
`k` is

`[k] = \left[(M\cdot L\cdot T^(-2))/(L)\right] = \boxed{[M\cdot T^(-2)]}`

2.

a. With N = Newtons, A = amperes, and m = meters,

`F = \frac{k I_1 I_2}d \iff \mathrm N = (k \,\mathrm A\,\mathrm A)/(\rm m) \implies k = \boxed{(\rm Nm)/(\rm A^2)}`

b. The dimension of
`k` is

`[k] = \left[((M\cdot L\cdot T^(-2))\cdot L)/(I^2)\right] = \boxed{[M\cdotL^2\cdot T^(-2)\cdot I^(-2)]}`

(where I = current, with "I" as in capital i)