Difficulty: Intermediate | Prerequisites: Vector calculus basics, Coulomb's Law, electric field concept
Tags: electrostatics, Gauss's Law, electric field, surface charge density, capacitors, dielectrics, RC circuits, charging by induction, coaxial cable, capacitance per unit length, energy density, electric potential energy
This set of notes covers the foundational layer of electrostatics: how charges create fields, how fields behave in symmetric geometries, and how energy is stored in those fields. It then extends into capacitors (how we store charge and energy in circuits) and the behaviour of RC circuits during charging. If you are comfortable with Coulomb's Law and the idea of an electric field, you are ready for this material. If not, start there first.
These topics appear in nearly every E&M final. Gauss's Law problems with spheres and infinite planes are the single most predictable exam question type, and capacitor circuits (series, parallel, dielectrics) are a close second.
Electric fields from symmetric charge distributions are found using Gauss's Law by choosing a Gaussian surface that exploits the symmetry. Capacitors store energy in the electric field between their plates; series and parallel combinations follow reciprocal and additive rules, respectively. RC circuits describe how capacitors charge and discharge exponentially through resistors, with a time constant tau = RC.
Electrostatic induction
The redistribution of charges on a conductor caused by a nearby (but not touching) charged object. In simple terms, a charged rod held near a neutral conductor pushes like charges away and pulls unlike charges closer, without any charge transfer by contact.
Grounding (earthing)
Connecting a conductor to an effectively infinite reservoir of charge (the Earth), allowing charge to flow in or out until the conductor reaches zero potential relative to the ground. Think of it as opening a valve to let excess charge escape.
Surface charge density (sigma)
Charge per unit area on a surface, measured in C/m². For an infinite plane with surface charge density sigma, the electric field on each side is sigma/(2 epsilon_0), directed away from the surface if sigma is positive.
Gauss's Law
The total electric flux through any closed surface equals the enclosed charge divided by epsilon_0. Formally: the surface integral of E dot dA = Q_enc / epsilon_0. In simple terms, the "flow" of electric field lines out of a closed surface tells you how much charge is inside.
Gaussian surface
An imaginary closed surface chosen so that the electric field is either constant and perpendicular to the surface, or parallel to it (contributing zero flux). The surface itself is a mathematical tool, not a physical object.
Capacitance (C)
The ratio of stored charge to voltage across a capacitor: C = Q / V, measured in farads (F). Think of it as a measure of how much charge a device can hold per volt applied.
Dielectric constant (kappa)
A dimensionless factor by which an insulating material increases the capacitance of a capacitor compared to vacuum. Inserting a dielectric with kappa = 2.0 doubles the capacitance.
Time constant (tau = RC)
The characteristic time for an RC circuit. After one time constant, a charging capacitor has reached about 63% of its maximum charge. After about 2.3 time constants, it reaches 90%.
Electric potential energy (U) of a charge distribution
The total work required to assemble a charge distribution by bringing charges in from infinity. For a uniformly charged sphere, this is found by integrating the energy density (1/2) epsilon_0 E² over all space.
Capacitance per unit length
For a coaxial cable or other extended geometry, the capacitance divided by the cable's length: C/L = 2 pi epsilon_0 / ln(b/a), where a and b are the inner and outer radii. This is a property of the geometry, not of the total length.
When a positively charged rod is brought near a neutral electroscope (without touching), negative charges in the electroscope are attracted toward the rod and positive charges are repelled away. If the electroscope is then grounded while the rod is still nearby, the repelled positive charges drain away to Earth, leaving excess negative charge. Removing the ground connection traps that negative charge. When the rod is finally taken away, the negative charge redistributes evenly across the electroscope, and the leaves repel because they both carry the same (negative) sign.
The key sequence: bring rod near, ground, remove ground, remove rod.
The final charge on the electroscope is opposite in sign to the rod's charge.
The leaves repel because they share the same charge, not because they are attracted to anything.
Each infinite plane produces a uniform electric field of magnitude sigma/(2 epsilon_0) on either side, directed away from a positive plane and toward a negative plane.
For two parallel planes at x = 0 (charge density +sigma) and x = d (charge density -2 sigma):
Between the planes (0 < x < d): both contributions point in the +x direction (away from the positive plane, toward the negative plane). The total field magnitude is sigma/(2 epsilon_0) + 2 sigma/(2 epsilon_0) = 3 sigma/(2 epsilon_0).
For x < 0: the +sigma plane pushes left (magnitude sigma/(2 epsilon_0)), the -2 sigma plane pushes right (magnitude 2 sigma/(2 epsilon_0)). The net field points in the +x direction with magnitude sigma/(2 epsilon_0).
For x > d: the +sigma plane pushes right, the -2 sigma plane pushes left. The net field points in the -x direction with magnitude sigma/(2 epsilon_0).
So the region where |E| = sigma/(2 epsilon_0) is both x < 0 and x > d.
A solid insulating sphere of radius R carries total charge Q distributed uniformly throughout its volume.
Inside the sphere (r < R): choose a spherical Gaussian surface of radius r. Only the charge within radius r is enclosed. Since the charge density is uniform, Q_enc = Q(r³/R³). Gauss's Law gives E(4 pi r²) = Q(r³/R³)/epsilon_0, so E = kQr/R³. The field increases linearly with r inside the sphere.
Outside the sphere (r > R): the entire charge Q is enclosed. E = kQ/r², identical to a point charge.
Potential inside the sphere (r < R):
The potential is found by integrating E from infinity to r. Since V = -(integral of E dr), and the field changes character at r = R, the integral splits into two parts. The result is a parabolic function of r: V(r) = (kQ/2R)(3 - r²/R²). This follows a parabolic decrease as r increases from the centre. It is not constant, not zero, and not linear.
To find the total energy stored in the electric field of a uniformly charged insulating sphere:
Use the energy density u_E = (1/2) epsilon_0 E² and integrate over all space.
Split into two regions: inside (r < R) where E = kQr/R³, and outside (r > R) where E = kQ/r².
The inside integral gives (1/2) epsilon_0 integral from 0 to R of (kQr/R³)² times 4 pi r² dr.
The outside integral gives (1/2) epsilon_0 integral from R to infinity of (kQ/r²)² times 4 pi r² dr.
The total result is U = 3kQ²/(5R), or equivalently U = 3Q²/(20 pi epsilon_0 R).
A coaxial cable has an inner conductor of radius a (charge +Q) and an outer shell of radius b (charge -Q):
r < a (inside the inner conductor): E = 0 (conductor in electrostatic equilibrium).
a < r < b (between conductors): Gaussian cylinder of radius r and length L encloses charge Q. E = Q/(2 pi epsilon_0 r L), directed radially outward.
r > b (outside the outer shell): enclosed charge is +Q - Q = 0, so E = 0.
The potential difference between the conductors is V_a - V_b = (Q / 2 pi epsilon_0 L) ln(b/a).
Capacitance per unit length: C/L = 2 pi epsilon_0 / ln(b/a).
Parallel: capacitances add directly. C_parallel = C_1 + C_2.
Series: reciprocals add. 1/C_series = 1/C_1 + 1/C_2.
For C_1 = 3 muF and C_2 = 6 muF in parallel: C_12 = 9 muF.
This combination in series with C_3 = 9 muF: 1/C_total = 1/9 + 1/9 = 2/9, so C_total = 4.5 muF.
Voltage across C_3:
The charge on capacitors in series is the same. Q = C_total times V_battery = 4.5 muF times 12 V = 54 muC. The voltage across C_3 = Q/C_3 = 54/9 = 6 V.
When a dielectric (kappa = 2.0) is inserted into C_1 while the battery remains connected:
C_1 increases to kappa times C_1 = 6 muF.
The parallel combination C_12 becomes 6 + 6 = 12 muF.
The new total capacitance is 1/(1/12 + 1/9) = 108/21 ≈ 5.14 muF.
The total charge increases: Q = C_total times 12 V ≈ 61.7 muC.
Since C_3 is in series, the charge on C_3 equals the total charge, which has increased.
So the charge on C_3 increases.
For a 10 muF capacitor charged through a 2 M Omega resistor by a 12 V battery:
Time constant: tau = RC = (2 times 10⁶)(10 times 10⁻⁶) = 20 s.
Charge as a function of time: Q(t) = Q_max(1 - e^(-t/tau)), where Q_max = CV = 120 muC.
To reach 90% of maximum charge: 0.9 = 1 - e^(-t/tau), so e^(-t/tau) = 0.1, giving t = tau times ln(10) ≈ 20 times 2.303 ≈ 46.1 s.
Current at that time: I(t) = (V/R) e^(-t/tau) = (12/2 times 10⁶)(0.1) = 0.6 muA.
Gauss's Law: integral of E dot dA = Q_enc / epsilon_0
E from infinite plane: E = sigma / (2 epsilon_0)
E inside uniformly charged sphere: E = kQr / R³
E outside uniformly charged sphere: E = kQ / r²
V inside uniformly charged sphere: V(r) = (kQ / 2R)(3 - r²/R²)
Total energy of uniformly charged sphere: U = 3kQ² / (5R)
Capacitors in parallel: C_total = C_1 + C_2 + ...
Capacitors in series: 1/C_total = 1/C_1 + 1/C_2 + ...
Capacitance with dielectric: C' = kappa times C
Coaxial cable capacitance per unit length: C/L = 2 pi epsilon_0 / ln(b/a)
RC time constant: tau = RC
RC charging: Q(t) = Q_max(1 - e^(-t/tau))
RC charging current: I(t) = (V/R) e^(-t/tau)
Coaxial cables are everywhere in electronics, from your TV aerial connection to the cabling inside network infrastructure. The capacitance per unit length determines signal propagation characteristics and impedance matching. The RC time constant governs everything from camera flash charging circuits to the debouncing of mechanical switches in keyboards.
Students often confuse "grounding while the rod is nearby" with "grounding after the rod is removed." The order matters enormously: grounding while the rod is nearby lets charge flow selectively, while grounding after the rod is gone simply neutralises the object.
Students frequently apply the formula for a point charge (E = kQ/r²) inside a uniformly charged sphere. Inside the sphere, only the enclosed charge matters, and E increases linearly with r.
A common error with capacitor circuits is to add capacitances in series directly (as with resistors). Series capacitors add as reciprocals.
When a dielectric is inserted with the battery connected, the voltage stays fixed and the charge changes. When inserted with the battery disconnected, the charge stays fixed and the voltage changes. Mixing these up is one of the most common exam errors.
⚠️ Gauss's Law problems with spherical symmetry (both inside and outside the sphere) are near-certain on any E&M final.
⚠️ The charging-by-induction sequence (bring near, ground, unground, remove rod) is a favourite conceptual question. Know the final sign of the charge and that the leaves repel.
⚠️ Capacitor circuits with dielectrics inserted while the battery is connected versus disconnected: the exam will test whether you know which quantity stays constant.
⚠️ The coaxial cable problem (three regions, potential difference, capacitance per unit length) is a classic long-answer question worth significant marks.
True or false: the electric field inside a uniformly charged insulating sphere is zero.
Fill in the blank: when two capacitors are in series, the ______ on each capacitor is the same.
True or false: inserting a dielectric into a capacitor always increases the voltage across it.
Fill in the blank: the time constant of an RC circuit is tau = ______.
True or false: the electric field outside a coaxial cable (beyond the outer shell) is zero when the inner and outer conductors carry equal and opposite charges.
Answers: 1. False (it increases linearly with r). 2. Charge. 3. False (only if the battery is disconnected; if connected, the voltage stays fixed). 4. RC. 5. True.
Q: A positively charged rod is brought near a neutral electroscope. The electroscope is grounded, then ungrounded, then the rod is removed. What is the final state?
A: The electroscope has a net negative charge and its leaves repel. Grounding while the rod is nearby allows positive charge to drain away; removing the ground traps the remaining negative charge.
Q: Two infinite parallel planes carry charge densities +sigma and -2 sigma. What is the electric field magnitude between them?
A: 3 sigma / (2 epsilon_0). Both planes contribute fields in the same direction in the region between them.
Q: What is the electric field inside a uniformly charged sphere at distance r from the centre (r < R)?
A: E = kQr/R³. By Gauss's Law, the enclosed charge scales as r³, giving a field proportional to r.
Q: C_1 = 3 muF and C_2 = 6 muF in parallel, in series with C_3 = 9 muF, across 12 V. What is the voltage across C_3?
A: 6 V. The total capacitance is 4.5 muF, total charge is 54 muC, and V across C_3 = 54/9 = 6 V.
Q: In an RC circuit (C = 10 muF, R = 2 M Omega, V = 12 V), how long does it take to reach 90% of maximum charge?
A: t = RC ln(10) = 20 times 2.303 ≈ 46.1 seconds.
The energy stored in the electric field of a charged sphere connects directly to the concept of self-energy in advanced electromagnetism, and is the electrostatic analogue of the energy stored in an inductor's magnetic field (covered in the circuits notes). Capacitance per unit length of a coaxial cable feeds into transmission line theory and impedance matching in later courses.
electrostatics, Gauss's Law, electric flux, surface charge density, sigma, epsilon_0, permittivity of free space, uniformly charged sphere, insulating sphere, Gaussian surface, spherical symmetry, electric potential inside sphere, parabolic potential, capacitor, capacitance, series capacitor, parallel capacitor, equivalent capacitance, dielectric, dielectric constant, kappa, RC circuit, time constant, tau, charging curve, exponential decay, coaxial cable, capacitance per unit length, electrostatic energy, energy density, self-energy, charging by induction, electroscope, grounding