Electricity and Magnetism

The transformer, shown in fig. 9 , outputs power at 415 V , along a copper cable 50 m in length, to an electrical machine. The total resistance of the copper conductor ( 100 m there and back) is $0.0493 \Omega$ at an operating temperature of $60^{\circ} \mathrm{C}$. The machine takes a current of 200 A .

a) What is the total power that the transformer is supplying to the machine and cable?
b) What is (i) the power loss in the cable, and (ii) what percentage is this of the total power supplied?
c) Often the conductor size is chosen not on the basis of the steady current required, but on the short circuit current that might occur. If in our wiring, a short circuit occurred at the machine end of the cable, a current of 6000 A could be expected. Explain why this current is significantly less than that calculated from the 415 V supply and the $0.0493 \Omega$ resistance of the cable.
d) If the circuit breaker produces a delay of 0.4 s before it breaks the circuit, calculate the heat energy generated in this short time interval. Assume that the resistance of the wire does not change significantly as it heats up.
e) The heat energy required to raise the temperature of 1 kg of copper by $1^{\circ} \mathrm{C}$ is called the specific heat capacity of copper. It has the value is $385 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}$. We can use a simple formula,

$$\text { heat energy supplied = mass } \mathbf{x} \text { specific heat capacity } \mathbf{x} \text { temperature rise }$$

in order to determine the temperature rise of the copper cable, assuming no heat loss to the surroundings.
Calculate
(i) The mass of copper in the 100 m of cable, given that its

$$\begin{aligned} \text { cross sectional area } & =50 \mathrm{~mm}^{2} \\ \text { density of copper } & =8960 \mathrm{~kg} / \mathrm{m}^{3} \end{aligned}$$

(ii) Calculate the final temperature of the cable after 0.4s, if its initial temperature is $60^{\circ} \mathrm{C}$.

A single uniform underground cable linking A to B, 50 km long, has a fault in it at distance $d \mathrm{~km}$ from end A . This is caused by a break in the insulation at X so that there is a flow of current through a fixed resistance $R$ into the ground. The ground can be taken to be a very low resistance conductor. Potential differences are all measured with respect to the ground, which is taken to be at 0 V .

In order to locate the fault, the following procedure is used. A potential difference of 200 V is applied to end A of the cable. End B is insulated from the ground, and it is measured to be at a potential of 40 V .
a) What is the potential at X ? Explain your reasoning.
b) What is
(i) the potential difference between A and X ?
(ii) the potential gradient along the cable from A to X (i.e. the volts/km)?
c) The potential applied to end A is now removed and A is insulated from the ground instead. The potential at end B is raised to 300 V , at which point the potential at A is measured to be 40 V .
(i) What is the potential at X now?
(ii) Having measured 40 V at end B initially, why is it that 40 V has also been required at end A for the second measurement?
d) What is the potential gradient along the cable from B to X ?
e) The potential gradient from A to X is equal to the potential gradient from B to X .
(i) Explain why this is true
(ii) From the two potential gradients that you obtained earlier, deduce the value of $d$.

a) The power dissipated as heat in a resistor in a circuit is given by $P=V I$. Show that this may also be expressed as $P=I^{2} R$ and $P=\frac{V^{2}}{R}$.
b) A student goes out to purchase an electric heater for his flat. The salesman says that, to get more heat, he should purchase a heater with a high resistance because $P=I^{2} R$, but the student thinks that a low resistance would be best, because $P=\frac{V^{2}}{R}$. Explain who is correct.
c) Copper is a better conductor than iron. Equal lengths of copper and iron wire, of the same diameter, are connected first in parallel, and then in series. A potential difference is applied across the ends of each arrangement in turn, and the p.d. is gradually increased from a small value until, in each case, one of the wires begins to glow. Explain this, and state which wire will glow first in each case.

d) A surge suppressor is a device for preventing sudden excessive flows of current in a circuit. It is made of a material whose conducting properties are such that the current flowing through it is directly proportional to the fourth power of the potential difference across it. If the suppressor dissipates energy at a rate of 6 W when the applied potential difference is 230 V , what is the power dissipated when the potential rises to 1200 V ?

An electric current flowing through a wire produces a magnetic field around the wire. Four wires carrying identical currents are shown placed at the corners of a square. The symbol $\oplus$ indicates a current flowing along the wire into the page, and the symbol $\odot$ indicates a current flowing along the wire pointing out of the page. In which of the diagrams is the magnetic field at the centre of the square greatest?

Diagram A: $\oplus$ $\odot$ / $\odot$ $\oplus$
Diagram B: $\oplus$ $\odot$ / $\oplus$ $\odot$
Diagram C: $\oplus$ $\oplus$ / $\odot$ $\odot$
Diagram D: $\odot$ $\odot$ / $\oplus$ $\oplus$

A combination of resistors shown below represents a pair of transmission lines with a fault in the insulation between them. The wires have a uniform resistance, but do not have the same resistance as each other. The following procedure is used to find the value of the resistance $R_{5}$.

A potential difference of 1.5 V is connected in turn across various points in the arrangement.
With 1.5 V applied across terminals AC a current of 37.5 mA flows
With 1.5 V applied across terminals BD a current of 25 mA flows
With 1.5 V applied across terminals AB a current of 30 mA flows
With 1.5 V applied across terminals CD a current of 15 mA flows
a) Write down four equations relating the potential difference, the resistor values and the currents.
b) Determine the value of resistor $R_{5}$.
c) If the ends C and D are connected together, what would be the resistance measured between A and B?
d) If the length AC (and also BD) is 60 metres of resistive wire, how far from A (or C) does the fault occur?

The resistance of a wire is proportional to its length and inversely proportional to its cross sectional area. The resistance of a wire of length $\ell$ and cross sectional area $A$ is given by $R=\frac{\rho \ell}{A}$ where $\rho$ is a constant which depends upon the material of the wire.

Some metals are ductile, which means that they can be drawn into long thin wires. In doing so, the volume $V$ remains constant whilst the length increases and the cross sectional area of the wire decreases.

A wire of length 32 m has a resistance of $2.7 \Omega$. We wish to calculate the resistance of a wire formed from the same volume of metal, but which has a length of 120 m instead. a) Write down the relation between $V, A$ and $\ell$. Obtain an expression to show how $R$ depends upon the length $\ell$ of the wire and its volume $V$. b) Rewrite the equation with the constants $\rho$ and $V$ on one side and the variables we are changing, $R$ and $\ell$, on the other. c) Calculate the resistance of the longer wire.

A student decides to calibrate a thermistor in order to measure variations in the temperature of the room. He connects a small bead sized thermistor across the terminals of a 5 V power supply and in series with a 1 A ammeter. The resistance of the thermistor is $120 \Omega$ at room temperature. a) Instead of showing small variations in room temperature, the thermistor is likely to go up in smoke. Explain why.

In the light of his experience, he decides to redesign his simple circuit as is shown in figure 3 below. He has a few values of resistor R to choose from; $5 \mathrm{k} \Omega, 500 \Omega, 50 \Omega$.

b) State which value of $R$ would give the biggest variation of V with temperature. Explain your choice. c) State which value of $R$ would be most likely to cause the same problem as in (a). Again, explain your choice.

A simple circuit is set up as shown below. A resistor, which is an ohmic device, is connected to a variable power supply, and a current $I$ flows through it. The power supply is adjusted so that the potential difference in volts measured across the resistor varies as $V=4+2 t$ where $t$ is the time measured in seconds from the moment the supply is turned on.

a) Why does the equation for $V$ only work when $t$ is measured in seconds? b) What does the term ohmic mean for the resistor? c) What would be the potential difference across the resistor at the end of 6 seconds? d) If $R=\frac{V}{I}$, and the resistance of the resistor is $8.0 \Omega$, i. What current would be flowing after 6 seconds? ii. Write down the equation for the current after time $t$. e) Sketch a graph of the current against time for the first 6 seconds. f) From the graph or otherwise, calculate how much charge flows through the resistor in the first 6 seconds. g) Sketch a graph of the power against time for the resistor for the first 6 seconds. On the graph what feature corresponds to the energy dissipated in the resistor? h) The resistor will burn out when the power dissipated reaches 50 W. For how many seconds will the resistor survive in the circuit?

A circuit with three resistors in the form of a "T" arrangement is shown in figure 2 below. The resistor $R_{2}$ is not accessible and has to be determined by measurements between points A & B and then C & D. (the voltmeter is an ideal voltmeter with infinite resistance).

Measurement 1: 8.0 V is applied to AB and 5.0 V is measured between C and D . The current from the supply is 0.50 A .

The supply and the voltmeter are now interchanged for measurement 2 .
Measurement 2: 10 V is applied to CD and 4.0 V is measured between A and B .
a) For each of the measurements, sketch the circuit, and mark on the all information given, including the voltage values given and the paths of the current flow.
b) Write down either one or two simple equations for each circuit, involving the current, voltages and the relevant resistances.
c) Solve these equations to determine the values of $R_{1}, R_{2}$ and $R_{3}$.

The resistance of a wire is proportional to its length $l$, and inversely proportional to its cross sectional area $A$. The constant of proportionality, $\rho$ is known as the resistivity of the material.

$$R=\frac{\rho l}{A}$$

For copper, the resistivity $\rho_{\mathrm{cu}}=1.68 \times 10^{-8} \Omega \mathrm{~m}$ and for silicon, a semiconductor, with trace amounts of impurities, $\rho_{\mathrm{si}}=0.53 \Omega \mathrm{~m}$.
a) The resistance between two opposing faces of a copper cube of length $l=1 \mathrm{~m}$ and cross sectional area $A=1 \mathrm{~m}^{2}$ is $1.68 \times 10^{-8} \Omega$. What would be the length of the side of a cube of silicon with a resistance of $1.68 \times 10^{-8} \Omega$ between opposite faces?
b) Why does silicon have a resistivity $10^{8}$ times greater than copper?

A long coil, called a solenoid, is formed by winding a long piece of insulated copper wire of circular cross section and resistivity $\rho_{\mathrm{cu}}$, in a single layer, around a tube of length $L$, which also has a circular cross section. The radius of the wire is $r$ and the radius of the tube is $a$ with $r \ll a$. There are $N$ turns on the coil, which are closely spaced and form a single layer on the tube. Neglect the thickness of the insulation on the wire.

The strength of the magnetic field in the solenoid, $B$, is proportional to the current $I$ flowing in the coil, and is given by

$$B=\mu_{o} \frac{N}{L} I$$

with $\mu_{\mathrm{o}}$ being a constant of proportionality.
When making a single layer solenoid to produce a large field strength $B$, we want to know if it is better to use many turns of long, thin wire or fewer turns of shorter, thicker wire. i.e. how does $B$ depend upon the radius of the wire, $r$.

A circuit is set up with a cell of emf $\mathcal{E}$ and a resistor $R$ in series with the solenoid. The resistance of the coil is $R_{\mathrm{c}}$.

c) Write down an expression for the current $I$ in the circuit in terms of $R, R_{\mathrm{c}}$ and $\mathcal{E}$.
d) Given that the wire is of radius $r$ and the solenoid is of radius $a$,
(i) how many turns of wire, $N$, can be wound on the solenoid of length $L$, and
(ii) what is the length $l$ of the wire used? (assume $a \gg r$)
e) Write down an expression for the resistance of the solenoid wire, $R_{\mathrm{c}}$.
f) Now using these terms, write down an expression for the field strength $B$ in terms of $\mu_{o}, \mathcal{E}, r, R, \rho, L, a$.

There are two situations to examine; the first is where $R \approx 0$ so that the current is determined by the wire forming the solenoid.
g) If $R \approx 0$ how does $B$ depend upon the radius $r$ of the wire used?

The second case is where the resistance $R$ dominates and the current in the circuit is determined by this resistance. i.e. $R \gg R_{c}$
h) Now write down how the field $B$ depends upon $r$.
i) Sketch graphs on the same set of axes showing how $B$ depends upon $r$ in these two cases.

Resistors have a very important role to play in circuits and here are four examples in which the characteristics can be put to use.
a) In the circuit shown below with the arrangement of resistors $R_{1}$ and $R_{2}$, what is the ratio of $\frac{R_{1}}{R_{2}}$ so that the power dissipated in $R_{1}$ is equal to the power dissipated in one of the resistors $R_{2}$.

b) An analogue ammeter, with a moving coil and a needle on a scale, has a full scale deflection when a current of 2.0 mA flows through the coil which has a resistance $5.0 \Omega$. To measure a current of 0.20 A in a circuit, the ammeter would require a low value resistor to be connected in parallel so that not all the current flows through the coil. What is the value of the resistor needed?
c) If the $2.0 \mathrm{~mA}, 5.0 \Omega$ moving coil meter is to be used as a voltmeter instead to measure 4.0 V f.s.d., what value series resistor would be used?
d) In Figure 5, if $V_{\text {out }}=-2.0 \mathrm{~V}$, mark clearly on Figure 5 the potentials of terminals $\mathrm{C}, \mathrm{D}, \mathrm{E}$ and F.

Two resistors $R_{1}$ and $R_{2}$ are connected to the amplifier, as in Figure 6. No current can flow in to terminal C on the amplifier. The potential $V_{p}$ at terminal C is very small i.e. $-V_{\text {out }} / 10^{6}=V_{p}$
e) Write down the currents flowing through resistors $R_{1}$ and $R_{2}$ in terms of the polarities across them. What approximation can now be made in order to find a simple expression for the ratio of $V_{\text {out }} / V_{\text {in }}$, which is the gain of the amplifier when controlled by the resistors.

A circuit consists of a battery of emf $\varepsilon=4.5 \mathrm{~V}$ and negligible internal resistance, connected in series with three resistors, $R_{1}, R_{2}, R_{3}$ of values $200 \Omega, 300 \Omega$ and $400 \Omega$ respectively. A digital voltmeter is connected between points A and B.

a) What is the potential difference between A and B measured by the voltmeter?
The voltmeter is replaced with an older moving coil meter, which itself has a resistance $R_{\mathrm{V}}$ of only $500 \Omega$.

b) What would be the potential difference between A and B measured by the voltmeter?

A circuit with two resistors, $R_{1}$ and $R_{2}$ connected in parallel, is shown in Fig 3 below. Current $I_{1}$ flows through $R_{1}$.

a) Obtain an expression for the total power, $P_{t}$, dissipated in $R_{1}$ and $R_{2}$, in terms of $R_{1}, R_{2}$ and $I_{1}$.
b) Sketch a graph of $P_{t}$ against $R_{2}$, as $R_{2}$ is varied from 0 to $\infty$. Mark on any values.

An ideal LED is connected in series with a resistor $R$ and a supply of emf $\varepsilon$ and zero internal resistance. The LED has forward conduction voltage $V_{c} = 1.8$ V.

a) If $\varepsilon = 5.0$ V and the resistor limits the current to 24 mA, what is the value of $R$?
b) Calculate the fraction of the power from the supply that is dissipated in the LED.
c) The supply is replaced with a sawtooth voltage with period 40 ms and peak 5.0 V (Fig. 6). Calculate (i) the time at which the LED first switches on, and (ii) the fraction of time for which the LED is lit.

d) Sketch a graph of the current through the LED against time, for two cycles.
e) Calculate the average power of the LED with the sawtooth emf.

A real LED has linear $I-V$ characteristic above $V_{c}$ with gradient $0.040 \mathrm{~A} \mathrm{~V}^{-1}$ (Fig. 7).
f) If the maximum current permitted is 10 mA, what would be the emf of the power supply?

(a) A battery is connected to a lamp, a moving coil ammeter (illustrated in Fig. 4) and a switch, all in series. The needle of the ammeter hits the end stop when the switch is first closed, but then returns to read the normal value for the particular lamp. A thermistor is then included in series in the circuit. The lamp again runs at almost normal brightness when the switch is closed, but the needle of the ammeter no longer hits the end stop. Explain why.

(b) A filament lamp has a resistance which we can assume is proportional to its temperature in kelvin. A 50 W bulb operates on 230 V at a temperature of 2250 K. What is the resistance of the bulb at room temperature of $27^{\circ} \mathrm{C}$?

A circuit of two resistors $R$ and $R_{\mathrm{C}}$ in series is connected to a supply as shown in Fig. 5. The potentials at three points are marked as $0 \mathrm{~V}, V_{\mathrm{A}}, V_{\mathrm{B}}$. The current $I$ in the circuit depends upon the value of $R_{\mathrm{C}}$.

(c) i. Obtain a relation between $V_{\mathrm{A}}, V_{\mathrm{B}}, I$ and $R$.
ii. Sketch a graph of the current $I$ ($y$-axis) against the potential $V_{\mathrm{B}}$ ($x$-axis). Mark on values where the line crosses each axis, and the gradient.
iii. $R_{\mathrm{C}}$ is now replaced by a filament light bulb. On your sketch graph above, add another line with an arrow that shows how $I$ and $V_{\mathrm{B}}$ vary from the moment the bulb is switched on until its steady illumination.

(d) A relay is an electro-mechanical device in which a small current flowing through a coil magnetically operates a switch. A relay is illustrated in Fig. 6. With no current in the coil, the switch is open. In the circuit shown in Fig. 7, a small current flowing through the coil will close the switch and light the bulb.

A potential divider circuit is made using a resistor, a cell and light dependent resistor (LDR). The resistor is in parallel with the coil.
i. In version A of the circuit, shown in Fig. 8, explain what would happen if light was shone on the LDR.
ii. In version B of the circuit, the LDR and resistor are interchanged. The bulb is now placed physically over the LDR, and the light bulb starts flashing on and off. The light of the bulb affects the LDR which switches the relay. Calculate the time taken for the filament to heat up to estimate the flashing frequency. Ignore any heat loss.

The bulb is $24 \mathrm{~W}, 12 \mathrm{~V}$ with a filament length of 12 cm. The temperature rise is from 300 K to 2300 K and the specific heat capacity of tungsten is $134 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$. The density of tungsten is $19300 \mathrm{~kg} \mathrm{~m}^{-3}$ and the resistivity can be taken as $66 \times 10^{-8} \Omega \mathrm{~m}$.

A linear electron accelerator consists of a series of hollow copper (drift) tubes of increasing lengths $\ell_{1}, \ell_{2}, \ell_{3}, \ldots$ along the beam and with a fixed small separation $d$ between each tube. The tubes are connected to a high voltage, constant radio frequency AC supply where the peak voltage of the AC is $V_{0}$. Adjacent tubes are connected so that they will always have opposite polarities, as shown in Fig. 9. When an electron of charge $e$ and mass $m_{e}$ is passing through the inside of a tube, its two ends are at the same potential and so the electron feels no force and is not accelerated. So it "drifts" through the tube. It passes through a large potential difference between the tubes and, if the charged particle's motion is in sync with the AC supply, when it leaves a tube the polarities have been reversed and the charge is accelerated into the next drift tube. A schematic diagram is shown in Fig. 10.

(a) i. Preliminary: a resistor has a potential difference of 5 V across it and a single electron flows through it. What is the thermal energy generated?
ii. To generate 2 W, how many electrons flow through in a second?

(b) i. Sketch a graph of the AC voltage against time for two cycles of the AC.
ii. What is the maximum potential difference between adjacent tubes connected to the AC supply as shown?
iii. The frequency of the AC is $f$. State in terms of $f$, the shortest time $T$ that the electron should take drifting through a tube in order that it experiences this maximum potential between tubes.

(c) i. The electrons leave the electron source in bunches starting with zero velocity, and are accelerated towards the first drift tube by the AC potential difference. In terms of $e, V_{0}$ and $m_{e}$, what is the maximum speed $v_{1}$ that the electron bunch can enter the first tube?
ii. Using your drift time from (b) iii and the speed from (c) i, obtain an expression for the length of the first drift tube, $\ell_{1}$ in terms of $e, V_{0}, m_{e}$ and $f$.

(d) i. The electron emerges from the end of the first tube with the same speed as it entered. What would be the speed of the electron as it leaves the second drift tube?
ii. What are the lengths of the drift tubes, $\ell_{2}, \ell_{3}$ in terms of the length of $\ell_{1}$?

(a) The terminal voltage of a dc power supply is measured as 5.00 V when it is on open circuit. A $2.00 \Omega$ resistor is connected across the terminals and the voltage drops by 0.100 V.
i. If the supply is treated as a simple emf and internal resistance, what would be the value of the internal resistance?
ii. If the load resistor is reduced to $0.400 \Omega$, what would be the terminal voltage now?

(b) Some electronic devices are designed to take a constant current, irrespective of the voltage applied.

Such a circuit with a constant current device attached is illustrated in Fig. 8, in which the constant current device takes 0.40 mA. A moving coil voltmeter with a needle moving over a scale, such as that illustrated on the right in Fig. 7, in fact works by taking a small sample of current and is factory calibrated to show a voltage on the scale. At full scale deflection, the meter draws a current of 1.0 mA. On the 300 V range, when connected in the circuit of Fig. 8, it reads 90 V.

What is the voltage applied to the device with the voltmeter removed from the circuit?

It is estimated that the average electric charge transported in a lightning strike is 20 C, and the energy converted is $2.4 \times 10^{10} \mathrm{~J}$.

(a) (i). What is the potential difference between the cloud and the ground, at the moment of the lightning strike?
(ii). The air breaks down and becomes conducting when the voltage across it exceeds $1.0 \times 10^{4} \mathrm{~V} \mathrm{~cm}^{-1}$. What would be the maximum height of the charged thundercloud above the ground so that a lightning strike takes place?

(b) In a typical thunderstorm lightning flashes strike the ground at intervals of 2 minutes. Over the whole surface of the Earth the total current carried in this way between the atmosphere and the ground averages 1200 A. Calculate the average number of thunderstorms taking place at any instant over the whole Earth.

(c) In order to determine whether these lightning strikes contribute much to the warming of the atmosphere, calculate the average heating power in a cubic metre of air due to these strikes if they all occurs in the lowest 1 km of the atmosphere.

Assume the Earth is a sphere of radius 6370 km.

An electric toaster, illustrated in Fig. 10, draws a constant 1100 W of power.

Mains electricity is supplied at 230 V.

When a toaster is turned on, the heating elements (the "red hot wires") very quickly get up to their operating temperature of $300^{\circ} \mathrm{C}$, from room temperature of $20^{\circ} \mathrm{C}$, in a time of 5.0 seconds.

The elements are made of Nichrome, with a specific heat capacity of $450 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$ and a density of $8.31 \mathrm{~g} \mathrm{~cm}^{-3}$.

(a) Calculate the mass of the heating elements.
(b) Calculate the resistance of the heating elements.
The elements are made of wire with a circular cross-section and a diameter of 1.0 mm.

(c) Calculate the electrical resistivity of Nichrome.
(d) In reality, the power drawn by the toaster will not quite be constant. Explain what you would expect to happen to the power during those 5.0 seconds, and why.

A false economy

"It's always cheaper to buy in bulk". An engineer takes this rather too literally, and insists on buying a large number of $4 \Omega$ resistors to use for all their circuits, instead of getting many different values.
(a) The engineer needs a $7 \Omega$ resistor for a circuit. Draw a circuit diagram showing how they could make this resistance out of their $4 \Omega$ resistors. (You're not allowed to 'cut up' any resistors!)
(b) It's good to be efficient. Most answers to (a) involve the use of more than 6 of the $4 \Omega$ resistors. If that applies to your solution, it's possible to do it with fewer. Try again, to design your $7 \Omega$ resistor using the fewest possible number of $4 \Omega$ resistors.