PhysicsCBSE Class 12 · Board exam 2023, set 55/1/1 · 3 marks · Short answer

CBSE Class 12 Physics 2023: question 27(a)

Set in the CBSE Class 12 board exam of 2023, paper 55/1/1, section C, question 27(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q27(a) · 3 marks · Short answerAlternating Current

How does the resistance differ from impedance ? With the help of a suitable phasor diagram, obtain an expression for impedance of a series LCR circuit, connected to a source $v = v_{m} sin ω t$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 55/1/1.

Hints

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

Hint 1

In a series circuit the current is the same through all elements — draw it as the reference phasor.

Hint 2

$V_{L}$ leads and $V_{C}$ lags the current by 90°.

Hint 3

Apply Pythagoras to the resultant of the phasors.

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.

Difference: resistance is the opposition offered by a resistor to both AC and DC and is independent of frequency, whereas impedance is the combined opposition of R, L and C to AC only and depends on the frequency of the source.
1 mark
With current $i = i_{m} sin ω t$ common to all elements: $V_{R} = i_{m} R$ (in phase with $i$ ), $V_{L} = i_{m} X_{L}$ (leads $i$ by $π /2$ ), $V_{C} = i_{m} X_{C}$ (lags $i$ by $π /2$ ). Phasor diagram (described): $V_{R}$ along the current phasor, $V_{L}$ and $V_{C}$ perpendicular to it in opposite directions; their resultant $(V_{C} - V_{L})$ combines with $V_{R}$ to give $V_{m} = V_{R} + V_{L} + V_{C}$ .
1 mark
From the right triangle of phasors: $V_{m}^{2} = V_{R}^{2} + (V_{C} - V_{L})^{2} = i_{m}^{2} [R^{2} + (X_{C} - X_{L})^{2}]$ , so $i_{m} = \frac{V _{m}}{R ^{2} + ( X _{C} - X _{L} ) ^{2}} = \frac{V _{m}}{Z}$ , where impedance $Z = R^{2} + (X_{C} - X_{L})^{2}$ .
1 mark

Answer: $Z = R^{2} + (X_{C} - X_{L})^{2}$

Where the marks are

Any one correct difference between resistance and impedance (1 mark)
Phasor diagram with $V_{R}$ , $V_{L}$ , $V_{C}$ and resultant (1 mark)
$V_{m}^{2} = V_{R}^{2} + (V_{C} - V_{L})^{2}$ leading to $Z = R^{2} + (X_{C} - X_{L})^{2}$ (1 mark)

Revise the chapter

Chapter: Alternating Current (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Alternating Current, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 55/1/1: every question, by section.

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CBSE Class 12 Physics 2023: question 27(a)

Hints

Step-by-step solution

Revise the chapter

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