Circuits

In a circuit the following resistor combination is found.

All the resistors in Figure have resistance $R$ ohms.

What is the total resistance across (i) AC and (ii) AB?

A battery of emf $E$ and internal resistance $r$ drives $3.0$ A round a circuit consisting of two $2.0 \mathrm{ohm}$ resistors in parallel. When these resistors are connected in series the current is $1.2 \mathrm{~A}$. Calculate $E, r$ and the power dissipated, $W$, in each resistor. [5]

A chain of resistors is composed of $n$ units, each consisting of three resistors, each resistor of resistance $R$. A unit is attached to the left hand end of the chain in order to increase the number of units from $n$ to $(n+1)$.

(i) Calculate the resistance (between $A$ and B) across a chain with two units, $R_{2}$, and the resistance $R_{3}$, across a chain with three units .

(ii) A unit is attached to a long chain. The resistance of the chain, $R_{T}$, is not altered by this addition. Determine the resistance of the chain. [6]

The circuit in Figure contains a cell of emf $E$, a known variable resistance $R_{0}$, an unknown resistance $R$ and an ammeter. When $\mathrm{X}$ and $\mathrm{Y}$ are short circuited $E=I_{0} R_{0}$ When $R$ is inserted the current is $\alpha I_{0}$, where $\alpha$ is a constant.

(i) Express $R$ in terms of $R_{0}$ and $\alpha$, giving the range of validity of $R$ and $\alpha$.

(ii) In order to extend the range of $\alpha$, modify the circuit by putting $R$ in parallel with $R_{0}$. Determine the ranges of $R$ and $\alpha$ for the modified circuit. [4]

Determine, in the Figure, the total resistances, $R_{\text {TBC }}$, across $B C, R_{\text {TBD }}$ across $B D$ and $R_{\text {TBA }}$ across $A B$. [6]

An a.c. voltmeter displays the rms value of the voltage, $V$, for a.c. signals and also for periodic signals that are not sinusoidal. What reading will it display if connected to a periodic voltage, period $4 T$, that changes instantaneously from $+10 \mathrm{~V}$ to $-2 \mathrm{~V}$ to $+4 \mathrm{~V}$ repeatedly, the voltages lasting, respectively, for $T, T$ and $2 T$ ?

Determine the following;

(i) the mean voltage i.e. $V_{\mathrm{m}}$.

(ii) the rms voltage i.e. $V_{\text {rms }}$.

(iii) the rms value of deviation from the mean, $\left(V-V_{\mathrm{m}}\right)$ which is $V_{\mathrm{rmsm}}$. [4]

A horizontal square wire loop of side $4.00 \mathrm{~cm}$ has a resistance, $R$, of $2.00 \times 10^{-3} \Omega$. The loop is situated in a vertical downward magnetic field of $0.700 \mathrm{~T}$. When the field is switched off, it decreases to zero, at a uniform rate, in $0.800 \mathrm{~s}$.

Determine:

(i) the induced current, $I$, and its direction in the loop.

(ii) the energy dissipated in the loop. [7]

A charge of $0.5 \times 10^{6} \mathrm{C}$ passes through a $12 \mathrm{~V}$ battery when the battery discharges. Assuming that the p.d. across the terminals remains constant, calculate the time for which it can supply $0.45 \mathrm{~kW}$. [2]

Draw a general resistive network diagram with:

(i) two resistors in series which are, in turn, in series with three resistors in parallel.

(ii) five resistors that are not in series or parallel, or in a combination of series and parallel arrangements.

Calculate the resistance in (i) and (ii) if all the resistors have resistance $R$. [5]

A cell, a resistor and an ammeter of negligible resistance are connected in series and a current of $0.80 \mathrm{~A}$ is observed to flow when the resistor has a value of $2.00 \Omega$. When a resistor of $5.00 \Omega$ is connected in parallel with the $2.00 \Omega$ resistor, the ammeter reading is $1.00 \mathrm{~A}$. [5]

Calculate the emf and the internal resistance of the cell.

A battery with an emf of $6 \mathrm{~V}$ can produce a maximum current of $3 \mathrm{~A}$. A resistor is connected to the terminals whose value is such that the power dissipated in it is a maximum. Calculate the maximum energy which can be dissipated in the external resistor in one minute. [4]

Two resistors and two cells are connected in the circuit shown in Figure. One cell has an e.m.f. of $2.0 \mathrm{~V}$ and an internal resistance of $1.0 \Omega$, the other an e.m.f. of $1.5 \mathrm{~V}$ and an internal resistance of $0.5 \Omega$. The resistors are connected in series and the point between them is at earth, i.e. zero potential. Calculate

(i) the current through the cells,

(ii) the potential difference across each cell, and

(iii) the potential, relative to earth, at points $A$ and $B$. [4]

Determine the current in the $6.0 \Omega$ resistor shown in the Fig. The cells have no internal resistance. [5]

(i). Five resistors, $R_{1} \ldots R_{5}$ are connected in a circuit between points $\mathbf{A}$ and $\mathbf{B}$, as in Figure. Resistors $R_{2}$ and $R_{4}$ can be changed in value to be connecting wires with $R=0$, finite values, or open circuit with $R=\infty$. Write down the values of $R_{2}$ and $R_{4}$ so that the network between $\mathbf{A}$ and $\mathbf{B}$ is equivalent to

i. three resistors in series,

ii. three resistors in parallel, and

iii. two identical resistors in parallel.

(ii). Figure shows a simple circuit with two cells and three resistors. If the current in the ammeter shown in the figure is $2.0 \mathrm{~A}$ determine the unknown e.m.f. $\varepsilon$, of the battery, assuming the batteries and ammeter have no internal resistance. [4]

(i). In Figure a, resistors $R_{1}, R_{2}$ and $R_{3}$ are connected between $\mathbf{A}$ and $\mathbf{B}$. Derive an expression for $R_{3}$ in terms of $R_{1}$ and $R_{2}$, if the equivalent resistance, $R_{A B}$ is equal to $R_{1}$.

(ii). A different arrangement is shown in Figure $\mathbf{b}$ for resistors $R_{1}, R_{2}$ and $R_{3}$. Again the equivalent resistance, $R_{A B}=R_{1}$, and the ratio $\frac{R_{3}}{R_{2}}=6$. Determine the value of the ratio of $\frac{R_{1}}{R_{2}}$. [4]

For the circuit shown, what should be the value of the variable resistor $R$ in order to minimise the power converted in the $10 \Omega$ resistor? [3]

In the arrangement of capacitors in Fig, calculate the charge stored on the $4.0 \mu \mathrm{F}$ capacitor. [3]

Resistance of a filament light bulb is given by $R=A+B P$ where $A$ and $B$ are constants, and $P$ is the power emitted by the bulb. When operating at $230 \mathrm{~V}$ the power emitted is $100 \mathrm{~W}$. When switched on from cold, the filament has a resistance value $1 / 5$ of its operating resistance, and consequently for the same mains voltage, the power is instantaneously five times greater than when at its normal operating power.

(i) Determine the values of A and B (if you wish, you may leave the answers as products of integer values given).

(ii) What would be the steady emitted power of the bulb if connected to a $210 \mathrm{~V}$ supply? [5]

In the circuit shown in Fig. 2, the three cells each supply an emf of $5.0 \mathrm{~V}$ and have an internal resistance of $5.0 \Omega$. The external resistors also each have a resistance of $5.0 \Omega$.

What arrangement of switches gives

(i) the maximum current,

(ii) the minimum non-zero current.

(iii) Determine the current in each case. [3]

The circuit of Fig consists of four resistors and a switch $\mathbf{S}$. When the $\mathbf{S}$ is open, the current flowing through the $5 \mathrm{k} \Omega$ resistor is $I_{0}$. When $\mathbf{S}$ is closed, the current flowing through the same resistor is $I_{\mathrm{c}}$. What is the ratio $\frac{I_{\mathrm{c}}}{I_{\mathrm{o}}}$, giving your answer as the ratio of two integers.

A capacitor of value $1.0 \mathrm{~F}$ discharges through a device whose resistance $R$ varies linearly with applied potential difference, $V$, so that $R=A V+B$, where $A$ and $B$ are constants. The resistance of the device has a value of $10.0 \Omega$ when $V=6.0 \mathrm{~V}$, and $4.0 \Omega$ when $V=0.06 \mathrm{~V}$.

The capacitor is initially charged to a potential of $6.0 \mathrm{~V}$.

Determine how long it takes for the capacitor to discharge to $1 \%$ of this initial value. [6]

A cell of emf $\varepsilon$ with an internal resistance $r$ is connected across two resistors $R_1$ and $R_2$ in parallel, as in Fig. 6. Resistor $R_2$ is variable, whilst $R_1$ is fixed.
Obtain expressions for
(i) the current $I$ flowing through the cell in terms of $\varepsilon, R_1, R_2, r$,
(ii) the current $I_2$ flowing through $R_2$ in terms of $I, R_1, R_2$,
(iii) the power $P_2$ converted in $R_2$ in terms of $\varepsilon, R_1, R_2, r$.
(iv) By considering the term $\frac{1}{P_2}$ or otherwise, determine an expression for $R_2$ in terms if $R_1$ and $r$ such that the power $P_2$ converted in $R_2$ is a maximum. [5]

Three capacitors, $C_1, C_2$ and $C_3$ are shown connected to a pair of cells of emfs $\varepsilon_1$ and $\varepsilon_2$ as shown in Fig. 7. Obtain an expression in terms of the capacitor values and the emfs for the potential across $C_3$. It is important that you write down your initial equations clearly. [5]

A circuit with an ideal diode in series with a resistor $R$ is shown in Fig. 2. Two supply voltages each with amplitude $V_0$ and frequency $f = \frac{1}{T}$ are also shown: in (a) as a square wave, and (b) a sinusoidal variation with time. Obtain expressions for the average power dissipated in the resistor $R$ for

• (i) the square wave signal shown in (a), and
• (ii) for the sinusoidal variation shown in (b).

A uniform wire of resistance per unit length $r$ is formed into a square of side $\ell$, with diagonals that are connected at the centre. This is shown in Fig. 3.

Obtain an expression for the resistance between points A and B at the corners of the square, in terms of $r$ and $\ell$. Give your answer in surds.

[3]

A length of resistance wire of diameter $1.4\,\text{mm}$ and resistivity $3.0 \times 10^{-4}\,\Omega\,\text{m}$ is bent to form a circle, as shown in Fig. 4. The radius of the circle is $10\,\text{cm}$. Two conducting wires are connected at points A and B, such that the contact B can slide around the circle.

1. (i) Calculate the maximum resistance that can exist between A and B.
2. (ii) Sketch a graph of the resistance between A and B against the arc length measured from A as contact B is slid from A around the circle and back to A on the other side.
3. (iii) What is the shortest length of the arc between A and B such that the resistance between A and B is $22.0\,\Omega$?

[6]

A circuit consists of an ideal battery (negligible internal resistance) and three resistors. Two of the resistors labelled $R_A$ have the same resistance and are connected in series with the battery, whilst the third, $R_B$ is connected in parallel with the battery. If $R_B$ and one of the $R_A$ are swapped,

1. (i) Find the upper limit of the factor by which the power dissipated by the circuit can increase due to this swap.
2. (ii) If swapping resistors $R_A$ and $R_B$ halves the total dissipated power, what is the ratio $\dfrac{R_A}{R_B}$? [6]

A household filament bulb has a rating of $240\,\text{V}$, $60\,\text{W}$, whereas a car filament headlamp bulb is rated at $12\,\text{V}$, $60\,\text{W}$. The filaments of the bulbs are both tungsten, of a similar length, and operate at the same temperature in order to produce the same spectrum. Determine the ratio of the masses of the two filaments. [3]

There is a thin rectangular sheet $ABCD$ of uniform resistivity, where $AB = 2BC$, as shown in Fig. 3. Two measurements of its resistance using an Ohmmeter have been made. The first is made across the mid-point of $AB$ and the edge $CD$ (a metal band is connected along $CD$), giving value $R_1$. The second is made between $A$ and $B$, giving value $R_2$. Find the ratio $\dfrac{R_1}{R_2}$. Show your ideas through diagrams.

[3]

An infinite tree of identical resistors $R$ is constructed as in Fig. 4. Three resistors are each connected to pairs of identical resistors which themselves are each connected to pairs stretching out to infinity. If we have a current entering at the centre of the tree of resistors and the entire outer edge is grounded, what is the resistance of the tree?

[3]