FSc Part 1 Physics Chapter 9 Complete Guide Book — Electrostatics and Current Electricity | PunjabTest.com

Rub a balloon on your hair and it picks up paper bits. That is electrostatics: charges sitting still, pulling and pushing. Let the charges move through a wire instead, and you have an electric current. This chapter covers both halves, and it starts with one blunt fact — charge does not come in any size you like.

An electron carries −e and a proton carries +e, where e = 1.6 × 10⁻¹⁹ C. That is the smallest free charge ever found. Anything bigger is built by adding or removing whole electrons, so any charge q on a body is a whole-number multiple of e: q = Ne, with N an integer. You will never find a body carrying, say, half an electron of charge, and that single rule decides several exercise questions later on.

Electrostatics is the study of charges at rest; once they are in motion we call it current electricity. Keep that split in mind, because the first half of the chapter (Coulomb, field, flux, Gauss, potential) is all about charges that stay put, and the second half (current, resistance, power, Kirchhoff) is about charges on the move.

Charles de Coulomb measured the force between charged balls in 1784 (the book prints 1874 here, a slip — Coulomb died in 1806, and his torsion-balance work was in the 1780s). What he found is beautifully simple. Between two point charges q₁ and q₂ a distance r apart, the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them: F ∝ q₁q₂/r², or F = k q₁q₂/r².

Like charges repel, unlike charges attract, and the force always acts along the line joining the two charges. It is also mutual: q₁ pulls or pushes q₂ exactly as hard as q₂ pulls or pushes q₁, in the opposite direction — F₁₂ = −F₂₁. In free space k = 1/(4πε₀) = 9 × 10⁹ N m² C⁻², where ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻² is the permittivity of free space.

Put an insulator — a dielectric — between the charges and the force drops. A dielectric is rated by its relative permittivity ε_r: the force in the medium is F = (1/4πε₀ε_r) × q₁q₂/r². The book's Table 9.1 gives air 1.0006 (so close to 1 that we treat air as free space), glass 4.8–10, mica 3–7 and distilled water 78.5 — which is why charges in water barely feel each other compared with in air. Worked Example 9.1 in the book (three charges, +40 μC, +50 μC and −60 μC) resolves the forces into components and gets a resultant of 1975 N at 34° — the method, resolving one force into x- and y-components and adding, is exactly the method the numericals expect.

A charge does not have to touch another charge to push it. It sets up an electric field around itself, and any charge placed in that field feels a force. The strength of the field at a point is defined as the force on a unit positive charge placed there: E = F/q. It is a vector, pointing the way a positive test charge would be pushed, and its unit is the newton per coulomb (N C⁻¹), which is the same thing as volt per metre.

For a point charge Q, Coulomb's law gives E = (1/4πε₀) × Q/r² — same inverse-square fall as the force, because the field is just the force per unit charge. In vector form, E = (1/4πε₀)(Q/r²) r̂, where r̂ points away from a positive Q. And since F = qE, a charge q in a field E feels F = qE — the one-line relation behind half the numericals in this chapter.

Two worked examples show the two classic moves. Example 9.2: two positive charges, 16.0 μC and 4.0 μC, 3.0 m apart — between them the fields point in opposite directions, so they cancel at a point 1.0 m from the smaller charge (the book's algebra gives d = +1 m or −3 m, and the −3 m root is off to the side where the fields do not cancel, so it is rejected). Example 9.3: a proton whose electrostatic force equals its weight sits in a field E = mg/e = 1.0 × 10⁻⁷ N C⁻¹. Tiny field, because a proton weighs almost nothing.

Michael Faraday's idea was to draw the field. An electric field line is a curve whose tangent at any point gives the direction of E there, drawn so that the lines are crowded where the field is strong and spread out where it is weak. The number of lines per unit area, held perpendicular to the field, is proportional to the magnitude of E. That is the whole map-reading skill in one sentence.

Round a single positive charge the lines shoot radially outward; round a negative charge they dive radially inward. Put two identical positive charges side by side and the lines between them seem to push each other away — the middle region is a zero-field spot, a neutral zone. Put a positive and a negative charge together and the lines start on the positive and end on the negative, curving across the gap. Between the plates of a parallel-plate capacitor the lines run parallel and evenly spaced in the middle: a uniform field, the same strength everywhere.

Four properties to memorise, because the exercise asks for them: (1) lines start on positive charges and end on negative ones; (2) the tangent to a line gives E's direction; (3) lines are closer where the field is strong, farther apart where it is weak; (4) no two lines ever cross — E has only one direction at any point, and a crossing would give it two.

Flux is just a count: how many field lines pass through a given area. The book's picture makes it concrete — through area A, 4 lines pass; through the tilted area B, only 2. But counting lines is not a proper definition, so the book turns it into maths. Hold the area perpendicular to the field and the flux is φ_e = EA. Hold it parallel to the field and no lines cross it, so φ_e = 0.

In general the area is treated as a vector A, whose size is the area and whose direction is the normal (perpendicular) to the surface. If θ is the angle between E and that normal, the flux is φ_e = EA cos θ = E·A. Flux is a scalar — it is a dot product — and its SI unit is N m² C⁻¹. The trap students fall into is the angle: θ is measured from the normal to the surface, not from the surface itself. Numerical 9.6 turns on exactly that, so read its working carefully when you get there.

Gauss's law is the shortcut that saves you from adding up field contributions one by one. Take any closed surface, add up the flux through all its little patches, and the total is φ_e = Q/ε₀, where Q is the total charge enclosed. The shape of the surface does not matter, the size does not matter — only the charge trapped inside counts. Charges outside the surface contribute nothing net, because every line that enters must leave again.

The book proves it for a sphere of radius r around a point charge q: E is the same everywhere on the sphere and parallel to every patch's normal, so φ = E × 4πr² = (1/4πε₀)(q/r²) × 4πr² = q/ε₀. Then it stretches the sphere into an arbitrary blob (Fig. 9.14) and the flux stays the same — that is the point.

The payoff is the charged conducting sphere. All its charge sits on its surface, the field is radial and uniform over any concentric sphere of radius r > R, so E × 4πr² = q/ε₀ and E = (1/4πε₀) q/r² outside — exactly as if the whole charge were a point at the centre. Just outside the surface (r = R), E = (1/4πε₀) q/R². That is a learning objective in its own right, and it is why Gauss's law, not Coulomb's law, is the tool for symmetric charge distributions.

Potential is the bookkeeping of energy. Push a positive charge q from plate A to plate B against the field, keeping it in equilibrium (so it does not speed up), and the work you do becomes its potential energy: U_B − U_A = W_AB. Divide by the charge and you get the potential difference: ΔV = V_B − V_A = W_AB/q = ΔU/q. Its unit, joule per coulomb, is called the volt: 1 V = 1 J / 1 C.

Potential at a single point needs a zero reference, and the book takes infinity as zero: V = W/q, the work to bring a unit positive charge from infinity to that point. Both potential and potential difference are scalars — work and charge are both scalars, so nothing about direction survives into V.

Now the link the comprehensive question demands. Between two plates a distance d apart, the work done against the field gives V_B − V_A = −Ed, so E = −(V_B − V_A)/d = −ΔV/Δd. The minus sign is the physics: E points downhill, towards decreasing potential, and ΔV/Δd is the potential gradient — the steepest rate at which potential changes with distance. Example 9.4 uses it at once: plates 1.0 cm apart, 12 V across them, E = 12 / (1 × 10⁻²) = 1200 V m⁻¹. Example 9.5 lets an electron fall through 12 V and converts the lost potential energy into kinetic energy, getting v = 2.1 × 10⁶ m s⁻¹.

Joules are clumsy for atomic energies, so physicists made a smaller coin. Move an electron through 1 volt and it gains (1.6 × 10⁻¹⁹ C) × (1 V) = 1.6 × 10⁻¹⁹ J. That amount of energy is called one electron volt: 1 eV = 1.6 × 10⁻¹⁹ J. The definition in the book is worth quoting exactly: the energy acquired or lost by an electron in traversing a potential difference of one volt.

Example 9.6 shows why the unit is convenient: a particle with charge 2e falling through 3.0 V gains ΔU = qΔV = (2e)(3.0 V) = 6.0 eV = 9.6 × 10⁻¹⁹ J — one line in eV, a mess of exponents in joules. And note the trap the exercise sets: the electron volt is a unit of energy, not of potential difference, even though the word "volt" sits inside it.

Between two charged plates the field is uniform, so the force on a charge is the same everywhere: F = qE, and the acceleration is a = qE/m. A positive charge runs along the field towards the negative plate; a negative charge runs against it, towards the positive plate. With E = V/d, everything is computable from the voltage and the plate gap alone.

The book's worked case is worth doing slowly: V = 20 V, d = 2.0 cm, so E = 20 / (2 × 10⁻²) = 1000 V m⁻¹. For an electron, a = eE/m = (1.6 × 10⁻¹⁹ × 1000) / (9.1 × 10⁻³¹) = 1.76 × 10¹⁴ m s⁻² — an absurd acceleration by everyday standards, which is why electrons pick up enormous speeds in tiny gaps. Released from rest at the negative plate, it reaches the positive plate with 2aS = v², so v² = 2 × 1.76 × 10¹⁴ × 2 × 10⁻² = 7.04 × 10¹², and v = 2.65 × 10⁶ m s⁻¹. The same answer comes faster from energy (½mv² = eV), and in an exam either route earns the marks — but you must show which one you used.

Send a charged particle straight along the field and it goes straight. Send it in sideways — perpendicular to a uniform field — and it does something prettier. The field pulls it sideways with a constant force while its forward motion carries on untouched, exactly like a ball thrown horizontally in gravity. The result is a parabola.

The horizontal component of velocity stays constant because no force acts horizontally; the vertical component grows steadily under the electric force. A positive particle bends towards the negative plate, a negative particle towards the positive plate — Fig. 9.19 shows both cases. Numerical 9.5 is this section in exam form: the time in the field comes from the horizontal motion alone (t = length / horizontal speed), and the acceleration from a = qE/m, kept strictly separate.

In 1836 Michael Faraday built a hollow conductor and showed that the field inside it is zero. The reason is already in this chapter: free electrons in the metal shift until they cancel any field inside, and once equilibrium is reached, all excess charge sits on the outer surface. The conductor shields anything inside it from outside fields — and shields outside from anything inside, since the induced charges arrange themselves to kill the field within too.

Two consequences to note. First, field lines just outside a conductor meet its surface at right angles: any parallel component would drag free electrons along the surface, and at equilibrium no electron is moving, so no parallel component can survive. Second, only conductors can be Faraday cages — insulators have no free electrons to do the rearranging, so they cannot shield at all.

Daily life is full of cages you never notice. A car's metal body carries a lightning strike over its surface into the ground while the passengers inside feel nothing; an aeroplane's metal frame does the same in a storm; a microwave oven's metal body keeps the microwaves inside the oven instead of leaking into the kitchen. The book's learning objective also covers the converse shielding job: sensitive circuits are put in metal boxes so outside fields cannot interfere with them.

Current is the flow of charge, and in a metal wire the charge carriers are free electrons. Connect the wire to a battery and an electric field is set up at every point inside it at once. The electrons feel a force in the −E direction and start moving — but they do not accelerate freely. They keep bumping into the atoms of the lattice, handing their gained energy over, and the net result is a slow average crawl in the −E direction called the drift velocity, of the order of 10⁻⁴ m s⁻¹. That crawl is the current.

Two puzzles dissolve once you see this. A bulb lights the instant you flip the switch, even though individual electrons crawl: the field, and the push from electron to electron, propagates along the wire almost instantaneously, so every electron in the circuit starts drifting at once. And the definition is clean: if a net charge Q passes any cross-section in time t, the current is I = Q/t, in amperes — one ampere is one coulomb per second. Charge flowing one way all the time is direct current (D.C., what batteries give); charge reversing direction at regular intervals is alternating current (A.C., what power stations supply to homes).

One convention to fix before it confuses you. Early scientists believed current was positive charges flowing, and the convention stuck: conventional current is the hypothetical flow of positive charges, from the positive terminal towards the negative, from higher potential to lower. In a metal wire the real flow is electrons the other way, but every formula, field direction and potential in this book is defined for the positive test charge, so from here on "current I" always means conventional current.

Take a wire segment of length L and cross-section A. Its volume is AL. If n is the number of charge carriers per unit volume, the segment holds nAL carriers, and if each carries charge q, the total charge sitting in the segment is Q = nALq. Now let the carriers drift: the time for all of them to exit through one face is t = L/v, where v is the drift velocity. Current is charge over time, so I = Q/t = nALq / (L/v) = I = nAvq.

Read the formula as a story: more carriers (n), a fatter wire (A), faster drift (v) or bigger charge per carrier (q) all mean more current. In a metal the carriers are free electrons with q = −e, and the sign only tells you the drift direction; the magnitude of the current uses e. Example 9.7 puts numbers on it: a copper wire of area 2 × 10⁻⁶ m² carrying 3 A, with n = 8.5 × 10²⁸ m⁻³, gives v = I/(nAq) = 3 / [(2 × 10⁻⁶)(8.5 × 10²⁸)(1.6 × 10⁻¹⁹)] = 1.1 × 10⁻⁴ m s⁻¹ — about a tenth of a millimetre per second, which is why the word "drift" is honest.

Apply a potential difference V across a conductor and a current I flows. Ohm's law says that, provided the physical state (chiefly the temperature) does not change, the current is directly proportional to the voltage: I ∝ V, or V = IR. The constant R is the resistance of the conductor, and its unit is the ohm (Ω): a conductor has a resistance of 1 Ω if 1 A flows through it when 1 V is applied across its ends.

The condition matters as much as the formula. Heat a filament and its resistance changes, so the V–I graph bends and Ohm's law, in the strict proportional sense, stops applying to it. Short question 9.7 in the exercise plays on exactly that: a bulb's filament resistance is not a fixed number — it depends on how hot the filament is running, which is why the 500 W bulb and the 100 W bulb at the same voltage have very different filament resistances.

Resistance belongs to a particular wire; resistivity belongs to the material. Experiment says a wire's resistance is directly proportional to its length and inversely proportional to its cross-sectional area: R = ρL/A, where ρ (rho) is the resistivity, the resistance of a metre cube of the material. Its unit is ohm-metre (Ω m). Turn it around and ρ = RA/L. Conductance G = 1/R and conductivity σ = 1/ρ are the reciprocals, for people who prefer to talk about how easily current flows rather than how hard it is.

The book's Table 9.2 is worth a slow look: silver 1.52 × 10⁻⁸ Ω m and copper 1.69 × 10⁻⁸ Ω m are the best ordinary conductors — that is why electric wires are copper (silver would be better and absurdly expensive). Carbon sits at 3.5 × 10⁻⁵, germanium at 0.5 and silicon at 20–2300: the huge range is the whole reason electronics exists.

Heat a metal and its resistance rises. The lattice atoms vibrate with bigger amplitude, so they present a bigger target and collide with free electrons more often. Over a considerable range around 0 °C the change is nearly linear, and the fractional change per kelvin is the temperature coefficient α: α = (R_t − R₀)/(R₀t), and the same for resistivity, α = (ρ_t − ρ₀)/(ρ₀t). Some substances — germanium, silicon — go the other way: their resistance falls with temperature, so they have negative α. Example 9.8 (iron wire, 1.5 V, 0.75 A, 5.0 m, 2.5 × 10⁻⁷ m²) gives R = V/I = 2.0 Ω and ρ = RA/L = 1.0 × 10⁻⁷ Ω m. Example 9.9 (platinum, 10 Ω at 0 °C, 20 Ω at 193 °C) gives α = 10/(10 × 193) = 5.18 × 10⁻³ K⁻¹.

A battery lifts charge uphill through a potential difference V. Lifting charge Q costs work W = VQ, and power is work per unit time, so electrical power = VQ/t = P = VI. That is the general relation: whatever a source delivers at voltage V and current I, the power is their product, in watts.

In a resistor the supplied power is dissipated as heat, and substituting V = IR or I = V/R gives the two forms you will use most: P = I²R and P = V²/R. Which form to use is a tactical choice: if you know the current through the resistor, I²R is immediate; if you know the voltage across it, V²/R is immediate. Mixing them up — using the voltage across one component with the current through another — is the classic power mistake, and the numericals punish it.

Electromotive force sounds like a force and is not one. The emf E of a source is the energy supplied to unit charge by the cell: E = W/Q, in joules per coulomb, which is volts. It is the "cause": the cell keeps supplying this energy per coulomb whether or not any current flows. Potential difference is the "effect": what you actually measure across the terminals when current is flowing.

Why are they different? A real cell has its own internal resistance r, from the electrolyte between its electrodes — equivalent to a pure emf E in series with r. With an external resistance R, the current is I = E/(R + r), so E = IR + Ir. The term IR = V is the terminal potential difference; Ir is the drop wasted inside the cell itself. Switch open, no current, no Ir drop, and the voltmeter reads the full emf. Switch closed, current flows, and the terminal voltage falls short of the emf by Ir. Example 9.10: open-circuit 2.2 V, closed across 5.0 Ω it falls to 1.8 V, so I = 1.8/5.0 = 0.36 A and r = (2.2 − 1.8)/0.36 = 1.1 Ω.

When a circuit is too tangled for series-parallel reduction, Kirchhoff's two rules still work, because they are just conservation laws wearing circuit clothes. The first rule is conservation of charge: the sum of all currents meeting at a point is zero, ΣI = 0 — currents towards the point count positive, away count negative. In plain words, what flows into a junction must flow out: I₁ + I₂ = I₃ + I₄ for the book's four-wire junction. No charge piles up at a point, and no charge is created there.

The second rule is conservation of energy: the algebraic sum of potential changes around any closed loop is zero, ΣV = 0. Walk a loop and add up what you gain crossing cells and what you lose crossing resistors, and you must come back to where you started — the book's loop gives E₁ − IR₁ − E₂ − IR₂ = 0. The sign rules the book prints are the ones to drill: crossing a source from − to + is +E, from + to − is −E; crossing a resistor with the current is −IR, against it is +IR.

Example 9.11 is the full workout: three resistors (10 Ω, 30 Ω, 15 Ω) and three cells (40 V, 60 V, 50 V) in two loops, solved as loop currents I₁ = 0.66 A and I₂ = 0.22 A, giving 0.66 A through R₁, 0.44 A through the shared R₂ (the difference, in the direction of the larger), and 0.22 A through R₃. The book's procedure is worth copying step for step: draw the diagram, choose loops so every resistance appears at least once, assume loop currents all in the same sense, write the loop equations with the sign rules, then solve.

Four resistances R₁, R₂, R₃, R₄ wired as a diamond, a battery across one diagonal (A–C) and a sensitive galvanometer across the other (B–D): that is a Wheatstone bridge. Normally current flows through the galvanometer. But adjust the known resistances until the galvanometer reads zero, and something exact has happened: points B and D are at the same potential, so no current can flow between them.

Applying Kirchhoff's second rule to the two loops ABDA and BCDB with loop currents I₁ and I₂, and setting the galvanometer current I₁ − I₂ = 0 (so I₁ = I₂), the book derives the balance condition R₁/R₂ = R₃/R₄. That is the whole instrument in one fraction: with three resistances known and adjusted to balance, the fourth — the unknown — follows at once. The beauty is that the galvanometer's own resistance never enters the result, because at balance no current flows through it at all.

A voltmeter has a problem: to measure a voltage it must draw some current, and that current changes the very voltage it is measuring. Only an ideal voltmeter, with infinite resistance, would draw nothing. A potentiometer dodges the problem entirely by measuring at a null — a point where no current flows, so nothing is disturbed.

The device is a uniform wire AB of resistance R with a sliding contact C: as C moves from A to B, the resistance between A and C runs from 0 to R, and with a driver battery of emf E the potential drop across AC runs from 0 to E. Connect the unknown cell E_x between A and C through a galvanometer, slide C until the galvanometer shows no deflection, and at that null point E_x equals the drop across AC — E_x = E(ℓ/L), where ℓ is the balancing length AC and L the whole wire. For two cells the ratio of emfs is the ratio of balancing lengths: E₁/E₂ = ℓ₁/ℓ₂. One catch: the unknown emf must be smaller than the driver emf E, or no null point exists at all.

That is why the galvanometer's null method matters so generally. A galvanometer detects current, and in null methods it is used as a zero indicator, not a meter: adjust until it reads zero, and at that moment the potentials at its two ends are equal, no current passes through it, and its own resistance drops out of the answer. The book lists three advantages: the galvanometer's resistance cannot corrupt the reading, galvanometers detect currents as small as 10⁻⁹ A, and "no deflection" is a sharp, unmistakable balance condition. The same null idea sits behind both the Wheatstone bridge and the potentiometer.

A thermistor is a heat-sensitive resistor, made by heating semiconductor ceramic — mixtures of metallic oxides of manganese, nickel, cobalt, copper and iron — and pressing it into beads, rods or washers. Most thermistors have a negative temperature coefficient: their resistance falls as temperature rises. Positive-coefficient types exist too, but the negative kind does the famous jobs.

Why does the resistance fall? Heating gives more energy to the charge carriers (electrons or holes), so they move more freely — the opposite of a metal, where heating mainly makes the lattice vibrate and get in the way. That steep fall is what makes thermistors superb thermometers: with a high negative coefficient, a small temperature change near 10 K produces a big, easily measured resistance change, far more accurately than a metal thermometer could manage down there.

The book's applications are the ones to quote: temperature measurement in thermometers, air conditioners, refrigerators, heaters, microwave ovens and incubators; temperature compensation in oscillators, battery chargers and power supplies, where a thermistor cancels the drift that temperature would otherwise cause; inrush current limiting, softening the surge when a device first switches on; and use as a voltage divider — as a thermistor warms, its resistance falls, the voltage drop across it falls, and the voltage at the divider's midpoint rises, which can trip a circuit, for instance switching a fire alarm's NOT gate so its output goes high and the siren sounds.

A light dependent resistor (LDR), also called a photoresistor, is a resistor whose resistance falls as light intensity rises — the learning objective states it in one line, and the rest of the section is the mechanism. LDRs are typically cadmium sulphide deposited in a special pattern on an insulating plate. In darkness the semiconductor has few free electrons, so its resistance is high. Light photons hand energy to electrons in the outer orbits, freeing them to conduct; more light means more free electrons, which means lower resistance. Less light, fewer free electrons, higher resistance — and that resistance change is measurable, so it can drive a circuit.

The applications follow the mechanism. As light sensors, LDRs switch automatic lighting on at dusk and off at dawn, and set camera exposure from the available light. As a voltage divider, an LDR sits in series with a fixed resistor: in the dark its resistance is very high (the book's circuit uses a 100 kΩ standard), so most of the voltage drops across it; in light its resistance collapses and the voltmeter reading falls. Feed the midpoint into a NOT gate and the LDR becomes an ON–OFF switch for lighting.

The chapter closes with the same sensor idea poured into concrete. Carbon fibres conduct electricity, so inspectors embed them in a bridge slab during construction and join them into a conducting network. Pass a small current through the network and measure its resistance: while the bridge holds together the resistance stays the same, but if the concrete deteriorates or the fibres are damaged, the resistance rises — and sensors watching that rise can warn of cracks long before eyes can see them. Other methods help too: sensors that continuously monitor strain, vibration and temperature, and ultrasound for internal flaws like cracks or voids. Resistance, the quantity this whole second half has been about, ends the chapter as a diagnostic tool.