Question

Two wires x and y have the same resistivity but their cross sectional areas are in the ratio 2:3 and lengths in the ratio 1:2. they are first connected in series and then in parallel to a dc source. find the ratio of the drift speeds of electrons in the two wires for the two cases.

Answer 1

case i) Series
In series combination current I through each resistor will be the same
Current I= neAVd
n=number of electrons per unit volume
e=Charge of an electron
A=Area of cross section of wire
Vd= Drift speed

Vd= I / (neA)

Vd1= I / (neA1) --------------(1)
Vd2= I / (neA2) --------------(2)
Note that 'ne' is same for both the wires since the resistivity same means same material

From 1) and 2) Vd1 / Vd2 = A2 / A1 ie Ratio of drift speed is 3:2

case ii) Parallel
In Parallel combination potential 'v 'through each resistor will be the same
Current I= neAVd
By ohms Law v=IR
ie v= (neAVd) R
Vd= v / RneA and R=ΡL/A

R1=ΡL1/A1
R2=ΡL2/A2
Ρ=resistivity

Vd1= v / (R1 neA1)---------------(3)
Vd2= v / (R2 neA1)---------------(4)

From 3) and 4) Vd1 / Vd2 = L2/L1 ie Ratio of drift speed is 2 : 1

Answer 2

Answer:

The ratio of their drift speeds of the electrons in the two wires for the two cases 1. Series 3:2 ; 2. parallel 2:1

Explanation:

Given that:-

Ratio of cross-sectional areas= $A_{1}: A_{2}=2: 3$

Ratio of lengths = $l_{1}: l_{2}=1: 2$

Resistivity is same for both wires.

Current,$I= ne A V_{d}$

n= Number of electrons

e= Charge on the electron

A= area of cross section

vd=drift velocity

I. Series connection

$V_{d}=\frac{I}{n e A} \\\\ V_{d} \alpha \frac{1}{A}$

$\therefore & \frac{V_{d_{1}}}{V_{d_{2}}}=\frac{1 / A_{1}}{1 / A_{2}}=\frac{A_{2}}{A_{1}}=\frac{3}{2} \\\\ V_{d_{1}}: V_{d_{2}}=3: 2$

II. Parallel connection

$I=\frac{V}{R} \\\\=n e A V_{d} \\\\V_{d}=\frac{V}{R n e A}\\\\=\frac{V}{\frac{\rho l_{x} \times e A}{A}}\\\\=\frac{V}{\rho_{n e l}} \\\\V_{d} \alpha \frac{1}{l} \\\frac{V_{d_{1}}}{V_{d_{2}}}=\frac{1 / l_{1}}{1 / l_{2}}\\=\frac{l_{2}}{l_{1}}\\=\frac{2}{1} \\\\ V_{d_{1}: V_{d_{2}}}=2: 1$

The ratio of their drift speeds of the electrons in the two wires for the two cases. 1. Series 3:2 ; 2. parallel 2:1