In a metre bridge, the balance point is found at a distance l_{1} with resistances R and S as shown in the figure.An unknown resistance X is now connected in parallel to the resistance S and the balance point is found at a distance l_{2}. Obtain a formula for X in terms of l_{1}, l_{2} and S.

#### Solution

When resistance R and S are connected :

Since balance point is found at a distance *l*_{1} from the zero end,

R ∝ l_{1} and,

S ∝ (100 − *l*_{1} )

`therefore R/S=l_1/(100-l_1)`.............(i)

When unknown resistance X is connected in parallel to S , then the effective resistance in the right gap, `S'=(SX)/(S+X)` ...............(ii)

Now, the balance point is obtained at a distance l2 from the zero end,

R ∝ l2

S' ∝ (100−l2)

`therefore R/S'=l_2/(100-l_2)`.............(iii)

Substituting the value of *S*' from (ii),

`(R(S+X))/(SX)=l_2/(100-l_2)`............(iv)

Dividing equation (iv) by (i),

`(S+X)/X=l_2/(100-l_2)xx(100-l_1)/l_1`

`S/X+1=(l_2(100-l_1))/(l_1(100-l_2))`

`S/X=(l_2(100-l_1))/(l_1(100-l_2))-1`

`S/X=(100l_2-l_1l_2-100l_1+l_1l_2)/(l_1(100-l_2))`

`S/X=(100(l_2-l_1))/(l_1(100-l_2))`

`X=(l_1(100-l_2))/(100(l_2-l_1)) S`