Electric Charges and Fields: common doubts, answered
The questions students ask most often about Electric Charges and Fields, each with a short answer. For the full chapter, read the Electric Charges and Fields notes.
About the chapter
How does the electric field change with distance for a point charge, a line charge, a sheet and a dipole?
A point charge gives E ∝ 1/r², an infinite line charge E = λ/(2πε₀r) ∝ 1/r, an infinite sheet E = σ/(2ε₀) with no dependence on distance, and a dipole E ∝ 1/r³. Many questions ask what happens when the distance is doubled, and this pattern answers them at once: the field becomes one-quarter, one-half, unchanged and one-eighth respectively.
Electric charge, conductors and insulators
Read this section in the notes →Why does a glass rod become positive when rubbed with silk?
The rod loses electrons to the silk; no positive charge is added to it. Rubbing moves only electrons, because protons are locked inside nuclei and cannot be pulled out by friction. Electrons pass from the glass surface to the silk, so the silk ends up with an equal negative charge. The total charge of rod and silk together stays zero, which is conservation of charge at work.
Why can a charged comb attract small uncharged bits of paper?
The comb polarises the paper, and the nearer opposite charges are pulled harder than the farther like charges are pushed. A negatively charged comb drives some electrons in the paper slightly away, leaving the side facing the comb a little positive. Because the Coulomb force weakens with distance, the attraction on the near side beats the repulsion on the far side, so the paper feels a net pull although its total charge is zero.
What is the difference between a conductor and an insulator?
A conductor lets charge move freely through it, while an insulator keeps charge where it is placed. Metals contain electrons that are not tied to any one atom, so charge given to a metal spreads over it at once. In glass, plastic or rubber the electrons stay bound, so a charge stays at the spot where it was put. That is why a metal rod held in the bare hand cannot be charged by rubbing.
Basic properties of electric charge
Read this section in the notes →What does quantisation of charge mean, and why don't we notice it in daily life?
It means every charge is a whole-number multiple of the electron's charge: q = ne, with e = 1.6 × 10⁻¹⁹ C. Everyday charges of a microcoulomb contain about 10¹³ such units, so adding or removing a single electron changes the total by a fraction far too small to measure. At that scale charge behaves as if it were continuous, but when you count electrons you must use q = ne.
Is electric charge conserved even when particles are created or destroyed?
Yes. Particles can appear or disappear, but an isolated system always keeps the same net charge. When a neutron changes into a proton and an electron, the new charges +e and −e add to zero, matching the neutron's zero charge. No process has ever been seen to create or destroy net charge, so charge conservation is a reliable check on any reaction or charging process.
Coulomb's law
Read this section in the notes →Why is the constant in Coulomb's law written as 1/4πε₀?
It is a choice that makes later results, especially Gauss's law, simpler. The value is about 9 × 10⁹ N m² C⁻², with ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻², the permittivity of free space. The 4π belongs to the sphere of area 4πr² over which a point charge's field spreads; putting it into Coulomb's law means it drops out of the flux result φ = q/ε₀.
Does Coulomb's law work for charged bodies that are not point charges?
Strictly it applies to point charges, meaning bodies much smaller than the distance between them. For large bodies the separation is not clearly defined, and the charges on them can shift under each other's influence. You then split each body into small pieces, apply the law to every pair and add the forces as vectors. A uniformly charged sphere is a handy exception: outside, it acts like a point charge at its centre.
Forces between multiple charges
Read this section in the notes →How do I find the net force on a charge due to several other charges?
Find the force from each other charge on its own and add the forces as vectors; this is the superposition principle. Each pair interacts as if the rest were absent. Fix each direction from the signs, attraction or repulsion, resolve every force into x and y components, add the components, then find the magnitude and angle. Adding the magnitudes directly is the classic mistake and only works when all forces point the same way.
Electric field
Read this section in the notes →Why must the test charge be very small when defining electric field?
A small test charge does not disturb the charges that create the field. A large one brought near a charged conductor would push the conductor's charges around and change the very field being measured. So the field is defined as E = F/q in the limit q → 0. The field is a property of the source charges at that point, and it exists whether or not anything is placed there.
Electric field lines
Read this section in the notes →Why can two electric field lines never cross each other?
Because the field at any point has only one direction. The tangent to a field line at a point gives the direction of E there. If two lines crossed, the crossing point would have two tangents and so two field directions at once, which is impossible. For related reasons, electrostatic field lines start on positive charges, end on negative ones and never form closed loops.
What does the spacing of electric field lines tell us?
It shows the field's strength: lines crowded together mean a strong field, widely spaced lines a weak one. Around a point charge the same number of lines passes through every sphere centred on it, but the sphere's area grows as r², so the lines per unit area fall as 1/r². That matches the 1/r² fall of the field, so one picture gives both the direction and the relative size of E.
Electric flux
Read this section in the notes →Is electric flux a scalar or a vector quantity?
Electric flux is a scalar. It is the dot product of two vectors, Δφ = E·ΔS = E ΔS cos θ, where the area vector ΔS points along the normal to the surface. Flux can still be positive or negative: for a closed surface the normal is taken outward, so field lines leaving the surface give positive flux and lines entering it give negative flux. Its unit is N m² C⁻¹.
Electric dipole
Read this section in the notes →Which way does the electric dipole moment point, from positive to negative or negative to positive?
It points from the negative charge to the positive charge. The dipole moment p = q × 2a has a size equal to one charge times the separation, unit C m, and runs from −q to +q. This is opposite to the direction of the field lines between the two charges. On the dipole's axis the far field points along p, while on the equatorial plane it points opposite to p.
Why does the field of a dipole fall as 1/r³ instead of 1/r²?
Because the fields of its two equal and opposite charges nearly cancel far away. Each charge alone gives a 1/r² field, but from a great distance the two charges look almost like one point, so only a small difference between their fields survives. That difference carries an extra factor of separation over r, giving E ∝ p/r³. Doubling the distance therefore cuts a dipole's field to one-eighth.
Dipole in a uniform external field
Read this section in the notes →Why is the net force on a dipole zero in a uniform electric field?
The two charges feel equal and opposite forces, qE and −qE, so they cancel. These forces do not act along the same line, though, so they form a couple and give a torque τ = p × E, of size pE sin θ. The torque turns the dipole until p points along E. A net force on the dipole appears only in a non-uniform field, where its two ends feel fields of different strength.
Continuous charge distribution
Read this section in the notes →What is the difference between linear, surface and volume charge density?
They measure charge per unit length, per unit area and per unit volume. λ = ΔQ/Δl, in C m⁻¹, suits a thin wire or rod; σ = ΔQ/ΔS, in C m⁻², suits a sheet or the surface of a conductor; ρ = ΔQ/ΔV, in C m⁻³, suits a charged solid. Choose the one that matches the shape, then treat each small element as a point charge and add their fields.
Gauss's law
Read this section in the notes →Does Gauss's law hold for a closed surface of any shape and size?
Yes. The total flux through any closed surface equals q_enc/ε₀, whatever its size or shape. A larger surface around the same charge has a weaker field but a bigger area, and the two effects cancel exactly. Gauss's law is always true, but it helps you find E only when symmetry lets you choose a surface on which E is constant in size and simply related to the normal.
Do charges outside a Gaussian surface affect the electric field on that surface?
Yes, they change E at points on the surface, but they add nothing to the total flux through it. Every field line from an outside charge that enters the surface also leaves it, so its inward and outward flux cancel. In ∮E·dS = q_enc/ε₀, the E is the total field from all charges, inside and outside, while q_enc counts only the charge enclosed.
Applications of Gauss's law
Read this section in the notes →Why is the field of an infinite plane sheet of charge independent of distance?
Because the field lines from an infinite plane stay parallel and never spread out. Take a Gaussian cylinder piercing the sheet: flux leaves only through its two flat ends, so 2EA = σA/ε₀ and E = σ/(2ε₀), with no distance in the result. A real sheet is finite, so this works only at distances small compared with its size and away from its edges.
Why is the electric field zero inside a uniformly charged spherical shell?
A Gaussian sphere drawn inside the shell encloses no charge, and by symmetry the field would have the same size all over it, so E must be zero. Physically, the nearby part of the shell has less charge but is closer, while the far part has more charge but is farther away, and their effects cancel exactly. Outside the shell the field is the same as if all the charge sat at its centre.
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