Multiple Choice & Assertion-Reason

PHYExam Practice

This chapter-wide revision article pulls together the highest-yield relationships in Current Electricity: Ohm's law, resistivity and its geometry/temperature dependence, the drift model, cell equations, power, and the balance conditions of the Wheatstone bridge and potentiometer. ISC MCQs and Assertion-Reason items reward clean recall of SI units (e.g.

resistivity in $\Omega \cdot m$ ), scaling laws (stretching a wire), and conceptual reasons (why relaxation time falls with temperature). Treat every option as a units/scaling check first, then a concept check.

Ohm's law and resistance of a wire

V = IR, \qquad R = \dfrac{\rho \, l}{A}

V

potential difference (V),

I

current (A),

R

resistance (

\Omega

\rho

resistivity (

\Omega \cdot m

l

length (m),

A

area (

m^2

). Resistivity is a material property independent of

l

and

A

Stretching a wire (volume constant)

R' = R \left( \dfrac{l'}{l} \right)^{2}

When length is scaled by a factor

k

at constant volume,

A

falls by

k

R \propto l^2

; stretching to

k

times the length makes

R

become

k^2

times. Doubling length gives

4R

, tripling gives

9R

Drift velocity and current

I = n A e \, v_d, \qquad v_d = \dfrac{I}{n A e}

n

free-electron density (

m^{-3}

e=1.6\times10^{-19}

v_d

drift speed (m/s, typically

\sim 10^{-4}

m/s). The signal travels near light speed but electrons themselves drift slowly.

Terminal voltage and EMF

V = E - I r

E

EMF (V),

r

internal resistance (

\Omega

I

current drawn (A). Use

V = E + Ir

only while charging. At balance on a potentiometer the cell drives no current, so

V = E

Wheatstone balance and power

\dfrac{P}{Q} = \dfrac{R}{S}, \qquad P_{diss} = I^{2} R = \dfrac{V^{2}}{R} = VI

Bridge balanced when no galvanometer current flows. Power dissipated in

R

given current

I

or voltage

V

. Maximum power to external

R

occurs when

R=r

SI unit of resistivity is $\Omega \cdot m$ ; conductivity $\sigma = 1/\rho$ has unit $\Omega^{-1}\,m^{-1}$ (S/m). Resistance unit is $\Omega$ alone.
For three resistors in parallel use $1/R = \sum 1/R_i$ ; for $2,3,6\,\Omega$ the result is $1\,\Omega$ , which must be smaller than the smallest resistor.
A potentiometer is preferred over a voltmeter for EMF because at balance it draws no current, so there is no $Ir$ drop and it reads true EMF.
Resistance of a metal rises with temperature because thermal vibrations increase collision frequency, decreasing relaxation time $\tau$ ; both Assertion and Reason there are true and connected.
Maximum power transfer to the load occurs at $R=r$ , where efficiency is only $50\%$ — maximum power is not maximum efficiency.
Order-of-magnitude facts to memorise: $v_d \sim 10^{-4}$ m/s, $e=1.6\times10^{-19}$ C, copper $n \approx 8.5\times10^{28}\,m^{-3}$ .
Power forms $I^2R$ , $V^2/R$ and $VI$ are equivalent only through $V=IR$ ; pick the form using the quantity that is fixed (series share fixed $I$ , parallel share fixed $V$ ).

Where the marks go

Confusing the unit of resistivity ( $\Omega \cdot m$ ) with resistance ( $\Omega$ ) or with $\Omega/m$ .
Writing $R \propto l$ when stretching a wire and forgetting the area also changes, which gives $R \propto l^2$ at constant volume.
Using $V = E + Ir$ for a discharging cell — the correct discharge relation is $V = E - Ir$ .
Picking the $R=r$ option for maximum efficiency; $R=r$ maximises delivered power, not efficiency.

How the board asks it

Assertion–Reasonresistance rises with temperature via falling relaxation time
Assertion (A): The resistance of a metallic conductor increases with rise in temperature. Reason (R): With rise in temperature the relaxation time $\tau$ of free electrons decreases. Choose: (a) both A and R true and R is the correct explanation of A; (b) both true but R is not the correct explanation; (c) A true, R false; (d) A false, R true.
Assertion–Reasonpotentiometer draws no current at balance, so no $Ir$ drop
Assertion (A): A potentiometer is preferred over a voltmeter for measuring the EMF of a cell. Reason (R): At the balance point the potentiometer draws no current from the cell, so there is no $Ir$ drop. Choose the correct option among (a)-(d) as in the standard assertion-reason key.
Assertion–Reasonstretching a wire at constant volume gives $R \propto l^2$
Assertion (A): When a wire is stretched to twice its original length its resistance becomes four times. Reason (R): On stretching at constant volume the resistance varies as $R \propto l^2$ . Choose (a), (b), (c) or (d) as in the standard assertion-reason key.
Multiple choiceparallel combination $1/R = \sum 1/R_i$
Three resistors of $2\,\Omega$ , $3\,\Omega$ and $6\,\Omega$ are connected in parallel. The equivalent resistance is: (a) $11\,\Omega$ (b) $6\,\Omega$ (c) $1\,\Omega$ (d) $0.5\,\Omega$ . Select the correct option.
Multiple choiceSI unit of resistivity $\rho$ versus resistance
The SI unit of resistivity $\rho$ is: (a) $\Omega$ (b) $\Omega\,m$ (c) $\Omega\,m^{-1}$ (d) $\Omega^{-1}\,m^{-1}$ . State the correct option.
Multiple choicemaximum power transfer at $R = r$ with $50\%$ efficiency
Maximum power is delivered by a cell of internal resistance $r$ to an external resistor $R$ when: (a) $R = 0$ (b) $R = r$ (c) $R \gg r$ (d) $R = 2r$ . Choose the correct option.

PreviousWheatstone Bridge & Metre Bridge

Written for Sublevo. Question text quoted anywhere in these notes is the Council’s and carries its year and paper; the board’s own diagrams are not reproduced.