In a metre bridge, the balance point is found at a distance l1 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 l2. Obtain a formula for X in terms of l1, l2 and S.
Solution
When resistance R and S are connected :
Since balance point is found at a distance l1 from the zero end,
R ∝ l1 and,
S ∝ (100 − l1 )
`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`
