Electrostatics — MHT-CET Physics

Formulas (4)

• Coulomb's Law and Force Ratios · Coulomb's law
  $$F = \frac{1}{4\pi\varepsilon_0 K}\,\frac{q_1 q_2}{r^2}$$
• Where the Field or Force Is Zero on a Line · Unlike charges: distance beyond the smaller one
  $$\frac{|q_1|}{(L+x)^2} = \frac{|q_2|}{x^2} \;\Rightarrow\; x = \frac{L}{\sqrt{|q_1|/|q_2|} - 1}$$
• Superposition: Symmetric Arrangements and Field Lines · Field of a point charge
  $$\vec E = \frac{kq}{r^2}\,\hat r, \qquad \vec E_{\text{net}} = \sum_i \vec E_i$$
• A Charge Moving in, or Held by, a Uniform Field · Motion in a uniform field
  $$a = \frac{qE}{m}, \qquad \text{KE} = \frac{q^2E^2t^2}{2m}, \qquad qE = mg \text{ (balance)}$$

Watch out for (4)

• Adding the moved charge to one side only
  → Coulomb's Law and Force Ratios
• Looking between two unlike charges
  → Where the Field or Force Is Zero on a Line
• Zero field does not mean zero potential
  → Superposition: Symmetric Arrangements and Field Lines
• Using the electron's mass for a heavier particle
  → A Charge Moving in, or Held by, a Uniform Field

Formulas (2)

• Electric Flux and Gauss's Law · Gauss's law
  $$\oint \vec E\cdot d\vec A = \frac{q_{\text{enc}}}{\varepsilon_0}$$
• Fields of Spheres, Sheets, Cylinders and Conductors · Three results to recall
  $$E_{\text{sheet}} = \frac{\sigma}{2\varepsilon_0}, \qquad E_{\text{conductor}} = \frac{\sigma}{\varepsilon_0}, \qquad E_{\text{solid sphere}}(r \le R) = \frac{\rho r}{3\varepsilon_0}$$

Watch out for (3)

• Subtracting the wrong way round
  → Electric Flux and Gauss's Law
• Distance from the surface is not distance from the centre
  → Fields of Spheres, Sheets, Cylinders and Conductors
• Sheet versus conductor
  → Fields of Spheres, Sheets, Cylinders and Conductors

Formulas (2)

• Field and Potential on the Axis and the Equator · Short dipole
  $$E_{\text{axis}} = \frac{2kp}{r^3}, \quad E_{\text{equator}} = \frac{kp}{r^3}, \quad V_{\text{axis}} = \frac{kp}{r^2}$$
• Torque, Energy and Work for a Dipole in a Uniform Field · Dipole in a uniform field
  $$\tau = pE\sin\theta, \qquad U = -pE\cos\theta, \qquad W_{\theta_1 \to \theta_2} = pE(\cos\theta_1 - \cos\theta_2)$$

Watch out for (2)

• Field power versus potential power
  → Field and Potential on the Axis and the Equator
• Using sin θ for the work
  → Torque, Energy and Work for a Dipole in a Uniform Field

Formulas (5)

• Potential of Point Charges, and Equipotentials · Potential of charges
  $$V = \frac{1}{4\pi\varepsilon_0}\sum_i \frac{q_i}{r_i}$$
• Field From Potential, and Potential From Field · Field and potential
  $$E = -\frac{dV}{dx}, \qquad V_B - V_A = -\int_A^B E\,dx$$
• Potential Energy of a Group of Charges · Energy of a system
  $$U = \sum_{\text{pairs } i<j} \frac{kq_iq_j}{r_{ij}}$$
• Speed Gained Through a Potential Difference · Energy gained
  $$qV = \tfrac{1}{2}mv^2, \qquad v = \sqrt{\frac{2qV}{m}}$$
• Merging Drops and Connected Spheres · Merging drops
  $$V_{\text{big}} = n^{2/3}\,v$$

Watch out for (5)

• Dropping the sign of a negative charge
  → Potential of Point Charges, and Equipotentials
• Forgetting the minus sign
  → Field From Potential, and Potential From Field
• Missing a pair
  → Potential Energy of a Group of Charges
• Taking the charge ratio as the speed ratio
  → Speed Gained Through a Potential Difference
• Potential rising as n, or as the cube root of n
  → Merging Drops and Connected Spheres

Formulas (3)

• Capacitance of Plates and Spheres · Parallel-plate and spherical capacitors
  $$C = \frac{\varepsilon_0 A}{d}, \qquad C_{\text{sphere}} = 4\pi\varepsilon_0 R$$
• Series and Parallel Combinations · Combinations
  $$\frac{1}{C_s} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots, \qquad C_p = C_1 + C_2 + \cdots$$
• Reading Capacitance From a Graph · Slope of the line
  $$V = \frac{1}{C}\,Q$$

Watch out for (3)

• Half the voltage from ±V plates
  → Capacitance of Plates and Spheres
• Giving the bigger voltage to the bigger capacitor
  → Series and Parallel Combinations
• Steeper means bigger — on the wrong graph
  → Reading Capacitance From a Graph

Formulas (3)

• Battery Connected or Removed: What Changes · Full dielectric
  $$C = \frac{K\varepsilon_0 A}{d}; \qquad Q\text{ fixed: } U \to \frac{U}{K}; \qquad V\text{ fixed: } U \to KU$$
• A Slab Filling Part of the Gap · Partial slab
  $$C = \frac{\varepsilon_0 A}{d - t\left(1 - \dfrac{1}{K}\right)}$$
• Two Dielectrics: Series or Parallel From the Figure · Equal halves
  $$\text{stacked: } C = \frac{2K_1K_2}{K_1 + K_2}C_0, \qquad \text{side by side: } C = \frac{K_1 + K_2}{2}C_0$$

Watch out for (3)

• Not asking what is held fixed
  → Battery Connected or Removed: What Changes
• Subtracting t/K instead of t(1 − 1/K)
  → A Slab Filling Part of the Gap
• The same words, two different figures
  → Two Dielectrics: Series or Parallel From the Figure

Formulas (4)

• Three Forms of the Stored Energy · Stored energy
  $$U = \tfrac{1}{2}CV^2 = \frac{Q^2}{2C} = \tfrac{1}{2}QV, \qquad u = \tfrac{1}{2}\varepsilon_0E^2$$
• Energy of Series and Parallel Groups · Equal energy, series and parallel
  $$\tfrac{1}{2}C_sV_s^2 = \tfrac{1}{2}C_pV_p^2 \;\Rightarrow\; \frac{V_s}{V_p} = \sqrt{\frac{C_p}{C_s}}$$
• Joining Two Charged Capacitors: Common Potential and Energy Lost · Common potential and loss
  $$V = \frac{C_1V_1 + C_2V_2}{C_1 + C_2}, \qquad \Delta U = \frac{C_1C_2}{2(C_1 + C_2)}(V_1 - V_2)^2$$
• Pulling the Plates Apart: Work and Force · Isolated capacitor, gap multiplied by n
  $$W = (n - 1)\,\frac{\varepsilon_0AV^2}{2d}, \qquad F = \frac{Q^2}{2\varepsilon_0A}$$

Watch out for (4)

• Treating energy as proportional to charge
  → Three Forms of the Stored Energy
• Squaring the capacitance ratio
  → Energy of Series and Parallel Groups
• Energy conserved by assumption
  → Joining Two Charged Capacitors: Common Potential and Energy Lost
• Taking the final energy as the work
  → Pulling the Plates Apart: Work and Force