Question

A uniform conductor of resistance r is cut into n equal pieces. Half of them are joined in series and the remaining half of them are connected in parallel. If the two combinations are joined in series, the effective resistance of all the pieces is

Answer 1

first of all, we should know about resistance.it depends on which quantity?

according to formula of resistance in a wire of length l, cross sectional area A and resistivity of material is p,

R = pl/A ,

so, resistance is directly proportional to its length and inversely proportional to cross sectional area.

now if a wire of resistance r is cut into n equal pieces.

then, resistance of each piece, r' = r/n.

half are joined in series .

so, Req = r/n + r/n + r/n .... (n/2) terms

= r/2

and half are joined in parallel.

so, 1/R'eq = n/r + n/r + n/r + ... (n/2) terms

or, 1/R'eq = n²/2r

or, R'eq = 2r/n²

now, Req and R'eq are joined in series

so, R = Req + R'eq

= r/2 + 2r/n²

= r[ 1/2 + 2/n² ]

= r[ (n² + 4)/n² ]

= (n² + 4)r/2n² [this is required answer]