PhysicsCBSE Class 12 · Board exam 2024, set 55/1/1 · 3 marks · Short answer

CBSE Class 12 Physics 2024: question 23(a)

Set in the CBSE Class 12 board exam of 2024, paper 55/1/1, section C, question 23(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q23(a) · 3 marks · Short answerCurrent Electricity

Define 'current density'. Is it a scalar or a vector? An electric field $E$ is maintained in a metallic conductor. If n be the number of electrons (mass m, charge $- e$ ) per unit volume in the conductor and $τ$ its relaxation time, show that the current density $j = α E$ , where $α = (\frac{n e ^{2}}{m}) τ$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 55/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Current density is current divided by area. Think about which direction it has.

Hint 2

Find the drift velocity from the acceleration $e E / m$ acting for time τ.

Hint 3

Count the charge crossing area A in time Δt, then divide by A and Δt.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Current density at a point is the current flowing per unit area of cross-section held normal to the direction of flow: $j = I / A$ . Its SI unit is A m $^{- 2}$ .
½ mark
It is a vector quantity, directed along the flow of (positive) charge.
½ mark
An electron accelerates as $a = - e E / m$ . Averaging over the relaxation time τ gives the drift velocity $v_{d} = - \frac{e E}{m} τ$ . The charge crossing area A in time Δt is $I Δ t = n e A ∣ v_{d} ∣Δ t$ .
½ mark
Substituting $∣ v_{d} ∣ = \frac{e E τ}{m}$ : $I Δ t = \frac{e ^{2} A}{m} τ n Δ t ∣ E ∣$ , so $I = \frac{n e ^{2} τ}{m} A ∣ E ∣$ .
½ mark
Since $I = ∣ j ∣ A$ , we get $∣ j ∣ = \frac{n e ^{2}}{m} τ ∣ E ∣$ .
½ mark
The electrons drift opposite to $E$ and carry negative charge, so the current density points along $E$ . Hence $j = (\frac{n e ^{2}}{m} τ) E = α E$ .
½ mark

Answer: Current density = current per unit normal area (vector); $j = \frac{n e ^{2} τ}{m} E$

Where the marks are

Defines current density (½ mark)
States that it is a vector (½ mark)
Writes $I Δ t = n e A v_{d} Δ t$ with $v_{d} = e E τ / m$ (1 mark)
Uses $I = j A$ and concludes $j = (n e^{2} τ / m) E$ (1 mark)

Revise the chapter

Chapter: Current Electricity (notes, formulas, where marks are lost and every past question)
Topic: Drift Velocity, Mobility and Resistivity

Practise this chapterBoard-style questions from Current Electricity, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 55/1/1: every question, by section.

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CBSE Class 12 Physics 2024: question 23(a)

Hints

Step-by-step solution

Revise the chapter

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