A student measures the terminal potential difference `(V)` of a cell (of emf `epsilon` and internal resistance `r`) as a function of the current `(I)` flowing through it. The slope and intercept of the graph between `V` and `I`, then respectively, equal

Options

A. `-r and epsi`
B. `r and -epsi`
C. `-epsi and r`
D. `epsi and -r`

Answer

A

Explanation

Dependence of resistance on the length of a conductor is given by `R= rho (l)/(A)`
The change in length is very small so we shall proceed with differentiation method. But, first, we need to understand that a change in length is caused by extension and hence it will have its corresponding effect in the area of cross section. Yet we also know that the volume of the material can be assumed to be constant.
Let V be the volume of the conductor, then `V= Al rArr A = (V)/(l)`
Substituting the area of cross section in the original equation of resistance we get the following
`R = rho (l)/((V)/(l)) rArr R = (rho)/(V) l^(2)` ...(i)
Now differentiating above equation (i) we get the following:
`(dR)/(dl) = (rho)/(V) (2l) rArr dR =(rho)/(V) (2l) dl` ...(ii)
Dividing equation (ii) From equation (i) we get:
`(dR)/(R ) = ((rho)/(V) (2l) dl)/((rho)/(V) l^(2)) rArr (dR)/(R ) = 2 (dl)/(l)`
`rArr (dR)/(R ) xx 100 = 2(dl)/(l) xx 100`
The percentage change in resistance is almost double to that of the percentage change in length So, when length is increased by 0.1% then resistance will increases by 0.2%. Hence option (a) is correct. Note that the above result is valid only if changes are very small.