Electrostatics — JEE Mains Physics

Formulas (3)

• Coulomb's law and adding forces · Coulomb's law in vector form
  $$\vec F_{21} = \frac{kq_1q_2}{|\vec r_2 - \vec r_1|^{3}}\,(\vec r_2 - \vec r_1), \qquad F_{\text{medium}} = \frac{F}{K}$$
• Sharing charge by contact · Sharing and the largest force
  $$q' = \frac{q_1 + q_2}{2}, \qquad F \propto q(Q - q) \ \text{is largest at}\ q = \frac{Q}{2}$$
• Charged bodies at rest · Hanging balls and the liquid condition
  $$\tan\theta = \frac{F_e}{mg}, \qquad K = \frac{\rho}{\rho - \sigma}$$

Watch out for (9)

• The vector points from the source to the target
  → Coulomb's law and adding forces
• K divides the force; √K scales the distance
  → Coulomb's law and adding forces
• Forces add as vectors
  → Coulomb's law and adding forces
• Add the signs before halving
  → Sharing charge by contact
• Order of contact matters
  → Sharing charge by contact
• Equal sharing needs identical spheres
  → Sharing charge by contact
• Angle from the vertical, not between the threads
  → Charged bodies at rest
• A liquid changes two forces
  → Charged bodies at rest
• The net force on a ball at rest is zero
  → Charged bodies at rest

Formulas (3)

• Field of point charges and null points · Superposition and the null point of like charges
  $$\vec E = \sum_i \frac{kq_i}{r_i^{2}}\,\hat r_i, \qquad x_{\text{null}} = \frac{r}{1 + \sqrt{q_2/q_1}}$$
• Field of rods, arcs, rings and discs · Arc at its centre and ring on its axis
  $$E_{\text{arc}} = \frac{2k\lambda}{R}\sin\frac{\varphi}{2}, \qquad E_{\text{axis}} = \frac{kQz}{(z^{2} + R^{2})^{3/2}}$$
• Infinite sheets and line charges · Sheet and line
  $$E_{\text{sheet}} = \frac{\sigma}{2\varepsilon_0}, \qquad E_{\text{line}} = \frac{\lambda}{2\pi\varepsilon_0 r}$$

Watch out for (9)

• Unlike charges: never between them
  → Field of point charges and null points
• Add arrows, not numbers
  → Field of point charges and null points
• Field lines never cross
  → Field of point charges and null points
• The arc formula uses half the angle
  → Field of rods, arcs, rings and discs
• λ comes from the arc's own length
  → Field of rods, arcs, rings and discs
• Let symmetry cancel first
  → Field of rods, arcs, rings and discs
• One sheet is σ/2ε₀
  → Infinite sheets and line charges
• A sheet's field does not fall with distance
  → Infinite sheets and line charges
• Conducting plates rearrange their charge
  → Infinite sheets and line charges

Formulas (2)

• Flux through a surface and through a cube · Flux and Gauss's law
  $$\phi = \vec E \cdot \vec A, \qquad \phi_{\text{net}} = \oint \vec E \cdot d\vec A = \frac{q_{\text{enc}}}{\varepsilon_0}$$
• Gauss's law for spheres, shells and cylinders · Fields from Gauss's law
  $$E_{\text{sphere, in}} = \frac{\rho r}{3\varepsilon_0}, \qquad E_{\text{cylinder, in}} = \frac{\rho r}{2\varepsilon_0}, \qquad E_{\text{out}} = \frac{kQ}{r^{2}}$$

Reference tables (1)

Watch out for (9)

• Name the normal, not the plane
  → Flux through a surface and through a cube
• Only faces facing the field count
  → Flux through a surface and through a cube
• Inward flux is negative
  → Flux through a surface and through a cube
• A charge on the surface counts in part
  → Enclosed charge and sharing flux by symmetry
• Outside charges do not change the net flux
  → Enclosed charge and sharing flux by symmetry
• The flat face of a hemisphere
  → Enclosed charge and sharing flux by symmetry
• Inside a solid sphere the field grows
  → Gauss's law for spheres, shells and cylinders
• 'On the surface' of a shell
  → Gauss's law for spheres, shells and cylinders
• Varying density needs an integral
  → Gauss's law for spheres, shells and cylinders

Formulas (3)

• Potential of charges, rings and shells · Point charges and a shell
  $$V = \sum_i \frac{kq_i}{r_i}, \qquad V_{\text{shell}}(r) = \begin{cases} kQ/R, & r \le R \\ kQ/r, & r \ge R \end{cases}$$
• Drops that merge and spheres joined by a wire · Drops and joined spheres
  $$V_{\text{big}} = n^{2/3}V, \qquad \frac{q_1}{q_2} = \frac{R_1}{R_2}, \qquad \frac{\sigma_1}{\sigma_2} = \frac{E_1}{E_2} = \frac{R_2}{R_1}$$
• Field from potential, work and potential energy · Gradient, work and energy
  $$\vec E = -\nabla V, \qquad W_{\text{ext}} = q(V_B - V_A), \qquad U = \sum_{i<j} \frac{kq_iq_j}{r_{ij}}$$

Watch out for (9)

• Potential has no direction
  → Potential of charges, rings and shells
• Inside a charged shell V is not zero
  → Potential of charges, rings and shells
• Outer shells use their own radius
  → Potential of charges, rings and shells
• Volume is conserved, not radius
  → Drops that merge and spheres joined by a wire
• Joined spheres share potential, not charge
  → Drops that merge and spheres joined by a wire
• Less charge, stronger field
  → Drops that merge and spheres joined by a wire
• Work by the field or by an agent
  → Field from potential, work and potential energy
• Keep the minus sign in E = −∇V
  → Field from potential, work and potential energy
• Count each pair once
  → Field from potential, work and potential energy

Formulas (2)

• Dipole moment, field and potential · Short dipole
  $$E_{\text{axial}} = \frac{2kp}{r^{3}}, \qquad E_{\text{equatorial}} = \frac{kp}{r^{3}}, \qquad V = \frac{kp\cos\theta}{r^{2}}$$
• Dipole in an external field: torque, energy and work · Torque, energy and work
  $$\vec\tau = \vec p \times \vec E, \qquad U = -\vec p \cdot \vec E, \qquad W = pE(\cos\theta_1 - \cos\theta_2)$$

Watch out for (6)

• Axial is twice equatorial
  → Dipole moment, field and potential
• p points from − to +
  → Dipole moment, field and potential
• Net charge makes p depend on the origin
  → Dipole moment, field and potential
• Half a turn costs 2pE
  → Dipole in an external field: torque, energy and work
• The minimum energy is negative
  → Dipole in an external field: torque, energy and work
• Unequal masses turn about the centre of mass
  → Dipole in an external field: torque, energy and work

Formulas (3)

• Motion in a uniform field · Deflection between plates
  $$y = \frac{qEL^{2}}{2mv^{2}}, \qquad \tan\theta = \frac{qEL}{mv^{2}}$$
• Holding and swinging charged bodies · Balance and small oscillations
  $$qE = mg, \qquad \omega^{2} = \frac{4kQq_0}{ma^{3}}$$
• Orbits round a line charge · Orbit round a line charge
  $$mv^{2} = 2k\lambda q, \qquad T = 2\pi r\sqrt{\frac{m}{2k\lambda q}}$$

Watch out for (9)

• Convert the field's units
  → Motion in a uniform field
• Only one velocity part changes
  → Motion in a uniform field
• Electrons go against the field
  → Motion in a uniform field
• Which way must E point?
  → Holding and swinging charged bodies
• Densities in SI units
  → Holding and swinging charged bodies
• Not every direction is stable
  → Holding and swinging charged bodies
• A line charge's field is 2kλ/r
  → Orbits round a line charge
• Same speed, different period
  → Orbits round a line charge
• A cylinder acts through its charge per length
  → Orbits round a line charge

Formulas (2)

• Capacitance of plates and spheres · Plates and spheres
  $$C = \frac{K\varepsilon_0 A}{d}, \qquad C_{\text{sphere}} = 4\pi\varepsilon_0 R, \qquad C_{\text{spherical}} = \frac{4\pi\varepsilon_0 R_1R_2}{R_2 - R_1}$$
• Series, parallel and networks · Series and parallel
  $$\frac{1}{C_{s}} = \sum_i \frac{1}{C_i}, \qquad C_{p} = \sum_i C_i$$

Watch out for (6)

• C does not depend on Q or V
  → Capacitance of plates and spheres
• Hollow or solid, the same C
  → Capacitance of plates and spheres
• Unequal plate charges
  → Capacitance of plates and spheres
• Decide by the nodes, not by the drawing
  → Series, parallel and networks
• A shorted capacitor stores nothing
  → Series, parallel and networks
• Steady state means no capacitor current
  → Series, parallel and networks

Formulas (2)

• Slabs and layers across the gap · Slab across the gap
  $$C = \frac{\varepsilon_0 A}{d - t + \dfrac{t}{K}}, \qquad \frac{1}{C} = \sum_i \frac{t_i}{K_i\varepsilon_0 A}$$
• Dielectrics side by side and stair plates · Dielectrics side by side
  $$C = \frac{\varepsilon_0}{d}\sum_i K_iA_i$$

Watch out for (6)

• Across the gap means series
  → Slabs and layers across the gap
• A metal sheet removes its thickness
  → Slabs and layers across the gap
• Thickness matters, position does not
  → Slabs and layers across the gap
• Side by side means parallel
  → Dielectrics side by side and stair plates
• Each part uses its own area
  → Dielectrics side by side and stair plates
• Read the boundary
  → Dielectrics side by side and stair plates

Formulas (2)

• Stored energy: battery connected or removed · Stored energy
  $$U = \tfrac{1}{2}CV^{2} = \frac{Q^{2}}{2C}, \qquad V\ \text{fixed: } U \to KU, \qquad Q\ \text{fixed: } U \to \frac{U}{K}$$
• Charge sharing and common potential · Common potential and energy lost
  $$V = \frac{C_1V_1 + C_2V_2}{C_1 + C_2}, \qquad \Delta U = \frac{C_1C_2}{2(C_1 + C_2)}(V_1 - V_2)^{2}$$

Watch out for (6)

• Decide what is fixed first
  → Stored energy: battery connected or removed
• The battery supplies twice the gain
  → Stored energy: battery connected or removed
• Percentages are squared
  → Stored energy: battery connected or removed
• Unlike plates subtract
  → Charge sharing and common potential
• Charge is kept, energy is not
  → Charge sharing and common potential
• Each capacitor's share is CV
  → Charge sharing and common potential