Electrostatics notes

MDCAT Physics

Electrostatics deals with electric charges at rest and the forces, fields, potentials and energy associated with them. This chapter also covers capacitors, dielectrics, charging and discharging, electric flux, and the basic role of the grid in a cathode ray oscilloscope.

Electric Charge and Coulomb's Law

Electric charge is a fundamental property of matter. There are two types of charge: positive and negative. Like charges repel each other, while unlike charges attract each other. Charge is conserved, which means it cannot be created or destroyed in an isolated system.

Coulomb's law gives the electrostatic force between two point charges. The force acts along the line joining the centres of the charges. Its direction is repulsive for like charges and attractive for unlike charges.

Coulomb's law: F = k|q1q2|/r2.
In vacuum, k = 1/(4πε0) and its value is approximately 9.0 × 10^9 N m2 C-2.
The force is directly proportional to the product of the magnitudes of the charges.
The force is inversely proportional to the square of the distance between the charges.
In a material medium, F = k|q1q2|/(K r2), where K is the dielectric constant.
Water has a dielectric constant of about 78. Therefore, the force between charges in water is approximately 1/78 of the force in vacuum.
The electrostatic force obeys the principle of superposition. The net force is the vector sum of all individual forces.
Charge is quantised: q = ne, where n is an integer and e is the elementary charge.

Electric Field and Electric Field Intensity

An electric field is the region around a charged body in which another charge experiences an electric force. A static charge always produces an electric field. The field exists even when no test charge is placed in it.

Electric field intensity at a point is the force acting on a unit positive test charge placed at that point. It is a vector quantity, so it has both magnitude and direction.

Electric field intensity: E = F/q0.
The SI unit of electric field intensity is N C-1 or V m-1.
N/C and V/m are equivalent units of electric field intensity.
N V-1 is not a unit of electric field intensity. The unit N/V is not equivalent to N/C or V/m.
The direction of E is the direction of force on a positive test charge.
For a point charge, E = kQ/r2.
The field due to a positive charge is directed away from the charge, while the field due to a negative charge is directed towards it.
For several charges, the resultant electric field is the vector sum of the fields due to individual charges.

Electric Field Lines and Electric Flux

Electric field lines are imaginary lines used to represent the electric field. The tangent to a field line at any point gives the direction of the electric field at that point. The closeness of the lines represents the relative strength of the field.

For an isolated point charge, field lines are straight radial lines. In a system of more than one charge, field lines may be curved. Field lines never cross because the electric field cannot have two directions at the same point.

Electric field lines start from positive charges and end on negative charges, or extend to infinity if no opposite charge is present.
The number of field lines passing through a surface is related to the electric flux through that surface.
Electric flux is Φ = EA cos θ for a uniform electric field passing through a plane surface.
Here, θ is the angle between the electric field and the normal to the surface.
For a closed surface, Gauss's law is Φ = Qinside/ε0.
The electric flux through a sphere containing a charge at its centre depends only on the amount of charge enclosed, not on the radius of the sphere.
Flux is maximum when the field is perpendicular to the surface and zero when the field is parallel to the surface.
The SI unit of electric flux is N m2 C-1.

Electric Field Due to Common Charge Distributions

The electric field produced by a charge distribution depends on the geometry of the distribution. Symmetry helps in finding the field of spherical shells, conducting surfaces and infinite sheets.

For an infinite uniformly charged sheet, the electric field is uniform and does not depend on the distance from the sheet. Its direction is normal to the sheet.

The field intensity due to an infinite sheet of surface charge density σ is E = σ/(2ε0) on either side of a single sheet.
Between two oppositely charged, parallel infinite plates, the fields add and E = σ/ε0.
Outside two oppositely charged parallel plates, the ideal fields cancel and the resultant field is zero.
For an infinite sheet, E is directly proportional to σ. A 25% increase in σ produces a 25% increase in E.
Inside a conductor in electrostatic equilibrium, the electric field is zero.
Excess charge on a conductor resides on its outer surface.
The electric field just outside a charged conductor is normal to the surface.

Electric Potential and Potential Energy

Electric potential at a point is the work done per unit positive test charge in bringing it from infinity to that point. It is a scalar quantity. Potential difference is the work done per unit charge in moving a test charge between two points.

A positive charge moved against an electric field gains electric potential energy. A positive charge moving along the electric field loses potential energy. The change in potential energy is related to potential difference by ΔU = qΔV.

Electric potential: V = W/q.
The SI unit of potential is volt, where 1 V = 1 J C-1.
Potential due to a point charge is V = kQ/r.
Electric potential is a scalar, so potentials are added algebraically.
For a positive point charge, potential is positive. For a negative point charge, potential is negative relative to infinity.
Potential difference is ΔV = W/q.
The potential gradient is ΔV/Δr. For a uniform field, E = -ΔV/Δr.
The negative sign shows that electric field points from higher potential to lower potential.
Equipotential surfaces have the same potential at every point. No work is done in moving a charge along an equipotential surface.
The potential energy of two point charges is U = kq1q2/r.

Applications of Potential and Charge Arrangements

The potential energy of a system of charges is found by adding the potential energy of every pair of charges. The sign of each pair contribution depends on whether the charges are like or unlike.

For three charges -q, Q and -q placed at equal separation a in a straight line, the pair distances are a, a and 2a. The total potential energy is U = k[(-qQ/a) + (Q(-q)/a) + (q2/2a)]. Setting U equal to zero gives Q:q = 1:4.

Work done in moving a charge through a potential difference is W = qΔV.
For a charge of 2e moving through 3.0 V, the energy magnitude is W = 2e × 3.0 = 9.6 × 10^-19 J.
A conductor in electrostatic equilibrium has the same potential at all points within it and on its surface.
An uncharged conducting sphere placed near a point charge becomes polarised by electrostatic induction.
If the sphere is far from the point charge compared with its radius, its potential is approximately the potential of a point charge at the sphere's centre, with induced-charge effects considered according to the conductor arrangement.
Electric field is the negative potential gradient: E = -dV/dr in differential form.
A zero potential at a point does not necessarily mean that the electric field at that point is zero.

Capacitors and Dielectrics

A capacitor is a device used to store electric charge and electrical energy. It consists of two conductors separated by an insulating material or dielectric. The capacitance measures the charge stored per unit potential difference.

For a parallel plate capacitor, increasing plate area increases capacitance, while increasing plate separation decreases capacitance. A dielectric inserted between the plates increases capacitance because it reduces the effective electric field and potential difference for a given charge.

Capacitance: C = Q/V.
The SI unit of capacitance is farad, F.
For a parallel plate capacitor in vacuum or air, C = ε0A/d.
With a dielectric of dielectric constant K, C = Kε0A/d.
The dielectric constant is K = C/C0, where C0 is capacitance without the dielectric.
If capacitance changes from 40 F to 80 F, K = 80/40 = 2.
A dielectric increases capacitance by a factor K.
The presence of a dielectric between two charged particles reduces their electrostatic force by a factor K.
The energy stored in a capacitor is U = 1/2 CV2 = 1/2 QV = Q2/(2C).
The energy density in an electric field is u = 1/2 εE2.

Capacitor Combinations, Charging and Special Cases

Capacitors may be connected in parallel or in series. In parallel, each capacitor has the same potential difference. In series, each capacitor carries the same magnitude of charge.

When an uncharged capacitor is connected to a battery through a resistor, current is initially maximum. As the capacitor charges, current decreases and eventually becomes zero. The potential drop across the resistor is initially maximum and finally becomes zero.

For capacitors in parallel, Ceq = C1 + C2 + C3 + ... and the voltage is the same across each capacitor.
For capacitors in series, 1/Ceq = 1/C1 + 1/C2 + 1/C3 + ... and the charge is the same on each capacitor.
For an RC charging circuit, q = Q(1 - e^(-t/RC)) and V = V0(1 - e^(-t/RC)).
The time constant is τ = RC. After one time constant, a charging capacitor has approximately 63% of its final charge.
During charging, I = I0e^(-t/RC). The slope of a charge-time graph gives current because I = dq/dt.
If C = 30 μF, I = 10 mA and final V = 300 V, then Q = CV = 9.0 × 10^-3 C and t = Q/I = 0.9 s.
An isolated conducting plate inserted midway between capacitor plates creates two capacitors in series. If the original separation is d and the plate is very thin, each part has capacitance 2C and their series combination is C.
If a conducting plate is connected to one outer plate and occupies half the separation, the effective separation is d/2 and the capacitance becomes 2C. The result depends on the electrical connection and geometry.
The grid in a cathode ray oscilloscope controls the number of electrons reaching the screen, so it controls the brightness of the spot.

Key terms

Electrostatics: The study of electric charges at rest.
Coulomb's law: The law giving the electrostatic force between two point charges as proportional to q1q2 and inversely proportional to r2.
Electric field: The region around a charge in which another charge experiences electric force.
Electric field intensity: Force acting per unit positive test charge at a point.
Electric field line: An imaginary line whose tangent gives the direction of the electric field.
Electric flux: A measure of the electric field passing through a surface.
Gauss's law: The total electric flux through a closed surface equals the enclosed charge divided by ε0.
Electric potential: Work done per unit positive charge in bringing it from infinity to a point.
Potential difference: Work done per unit charge in moving a charge between two points.
Potential gradient: The change in potential per unit distance, represented by ΔV/Δr.
Electric potential energy: The energy possessed by a charge because of its position in an electric field.
Capacitor: A device consisting of two conductors separated by an insulator that stores charge and energy.
Capacitance: The charge stored per unit potential difference, given by C = Q/V.
Dielectric: An insulating material placed between capacitor plates.
Dielectric constant: The ratio of capacitance with a dielectric to capacitance without it.
Time constant: The quantity RC that determines the rate of charging or discharging of a capacitor.

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Physics shortcuts

Comparing distance and displacement

Distance equals the magnitude of displacement only when the particle travels along a straight path without reversing direction.

Check whether the path is straight and one-directional.
If yes, distance = |displacement|.
Example: A particle moves 5 m east in a straight line. Distance = 5 m and displacement magnitude = 5 m.

This shortcut does not apply to a curved path or to motion involving a change of direction.

Projectile range and components

For a projectile launched and landing at the same level, use R = u² sin 2θ/g. Resolve the initial velocity into horizontal and vertical components when needed.

Write ux = u cos θ and uy = u sin θ.
For the same launch and landing level, R = u² sin 2θ/g.
Example: u = 20 m/s, θ = 30°, g = 10 m/s². R = 400 sin 60°/10 = 34.6 m.

The range formula does not apply directly when the projectile lands at a different height.

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