Comprehensive Notes on Electrostatics and Coulomb's Law

Introduction to Electrostatics

  • \textbf{Definition of Electrostatics}: It is the study of stationary electric charges, as opposed to moving charges (current) which are studied in the context of electrical circuits.
  • \textbf{Grade Focus}: This material covers curriculum requirements for Grade 11 and 12 students, building on Grade 10 foundations.
  • \textbf{Section 1 Recap}: Focuses on definitions, study tips, charge types, conversion scales, and the behavior of electric fields around point charges.
  • \textbf{Section 2 Coulomb’s Law}: Investigates electrostatic force using Coulomb’s Law, proportionality, vector diagrams in one and two dimensions, and exam-style problem-solving.

Fundamental Definitions and Principles (Grade 10 Recap)

  • \textbf{Charge}: A physical property of matter that causes it to experience a force when placed in an electromagnetic field.
  • \textbf{Principle of Conservation of Charge}: The net charge of an isolated system remains constant during any physical process. An example included is when two charges make contact and then separate; the total charge before equals the total charge after.
  • \textbf{Principle of Charge Quantization}: All charges existing in the universe consist of an integer multiple of the charge on one electron. The elementary charge value is 1,6×10−19‡C1,6 \times 10^{-19}‡C.
  • \textbf{Polarization}: The partial or complete separation of positive and negative electric charges within a system, effectively creating a dipole.

Essential Study Tips and Mathematical Tools

  • \textbf{Vector Diagrams}: A thorough revision of vector diagrams is required. Students must identify forces acting on charges and draw free-body diagrams (FBDs).
  • \textbf{Newton’s Third Law}: Every force interaction in electrostatics follows this law (forces are equal in magnitude and opposite in direction).
  • \textbf{Trigonometry}: Sine, cosine, and tangent ratios are frequently used as electrostatic arrangements often form triangles.
  • \textbf{Pythagoras Theorem}: This is mandatory for calculating the resultant force when charges are positioned at right angles to one another.
  • \textbf{Final Formatting}: The net force (FNETF_{NET}) must be stated as a single numerical value in Newtons (N) with a specific direction at the end of every calculation.

Electric Charge Properties

  • \textbf{Nature of Charge}: All matter contains charges. Charge cannot be created or destroyed, only transferred.
  • \textbf{Methods of Charging}: Objects can be charged via friction, touch, or induction (polarization).
  • \textbf{Symbols and Units}:     - Symbol for charge: QQ     - SI unit of charge: Coulomb (C)
  • \textbf{Elementary Particles}:     - \textbf{Electron}: Negative charge (−-), valued at −1,6×10−19‡C-1,6 \times 10^{-19}‡C.     - \textbf{Proton}: Positive charge (+‡+‡), valued at +1,6×10−19‡C+1,6 \times 10^{-19}‡C.     - \textbf{Neutron}: Neutral charge (0‡C0‡C).

Conversion Scale for Coulombs

  • 1 milliCoulomb (mC) = 1×10−3‡C1 \times 10^{-3}‡C
  • 1 microCoulomb (ΑC) = 1×10−6‡C1 \times 10^{-6}‡C
  • 1 nanoCoulomb (nC) = 1×10−9‡C1 \times 10^{-9}‡C
  • 1 picoCoulomb (pC) = 1×10−12‡C1 \times 10^{-12}‡C

Charge Transfer and Quantization Calculations

  • \textbf{Calculating Charge After Contact}: When two identical metal spheres touch and separate, the charge on each sphere is determined by:     - Qnew=Q1+Q22Q_{new} = \frac{Q_1 + Q_2}{2}
  • \textbf{Direction of Electron Movement}: Electrons move from the more negative sphere (excess electrons) to the more positive sphere (electron deficiency).
  • \textbf{Calculating Number of Electrons Transferred (nn)}:     1. First, calculate the charge transferred (QtransQ_{trans}): Qtrans=QF−QIQ_{trans} = Q_F - Q_I     2. Use the formula: n=Qtransqen = \frac{Q_{trans}}{q_e}, where qeq_e is the charge of a single electron.
  • \textbf{Case Study}: Spheres R and S.     - Charge R: +8‡μC+8‡\mu C     - Charge S: −4‡μC-4‡\mu C     - Net charge after contact: +8+(−4)2=2‡μC\frac{+8 + (-4)}{2} = 2‡\mu C on each sphere.     - Purpose of wooden stands: They ensure charge does not leak to the ground (act as insulators).

Electric Fields Around Point Charges

  • \textbf{Single Point Charge Rules}:     - Shape must be radial.     - Field lines must touch the charge but not enter the sphere.     - Field lines must never touch or cross each other.     - Approximately 8 lines are required for diagrams.     - \textbf{Positive Charge}: Arrows point radially away.     - \textbf{Negative Charge}: Arrows point radially towards.
  • \textbf{Field Between Two Charges}:     - \textbf{Like Charges (Repulsion)}: Shapes show lines bending away from each other; horizontal lines between spheres are crucial.     - \textbf{Opposite Charges (Attraction)}: Lines flow from the positive charge to the negative charge.

Coulomb’s Law

  • \textbf{Definition}: The magnitude of the electrostatic force exerted by one stationary point charge (Q1Q_1) on another stationary point charge (Q2Q_2) is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance (rr) between them.
  • \textbf{Mathematical Formula}:     - FE=kQ1Q2r2F_E = \frac{k Q_1 Q_2}{r^2}
  • \textbf{Variables}:     - FEF_E: Electrostatic force in Newtons (N).     - Q1,Q2Q_1, Q_2: Charges in Coulombs (C).     - rr: Distance between centers in meters (m).     - kk: Coulomb's constant = 9,0×109‡N⋅m2⋅C−29,0 \times 10^9‡N \cdot m^2 \cdot C^{-2}.
  • \textbf{Mathematical Relationships}:     - \textbf{Force vs. Charge}: F∝Q1Q2F \propto Q_1 Q_2. Doubling both charges results in 4F4F. Halving one while quadrupling the other results in 2F2F.     - \textbf{Force vs. Distance}: F∝1r2F \propto \frac{1}{r^2}. Halving distance results in 4F4F. Reducing distance to a third results in 9F9F.

Worked Examples and Calculations

  • \textbf{Example 1: Calculating Force}     - Spheres at 73 cm apart (0,73‡m0,73‡m).     - Q1=+3,4‡μCQ_1 = +3,4‡\mu C, Q2=−2,6‡μCQ_2 = -2,6‡\mu C.     - Calculation: FE=(9,0×109)(3,4×10−6)(2,6×10−6)(0,73)2=0,149‡NF_E = \frac{(9,0 \times 10^9)(3,4 \times 10^{-6})(2,6 \times 10^{-6})}{(0,73)^2} = 0,149‡N.     - Note: Only positive magnitude values are used in the formula calculation.
  • \textbf{Example 2: Calculating Distance}     - Force = 9,9×10−5‡N9,9 \times 10^{-5}‡N, QX=+5,5‡nCQ_X = +5,5‡nC, QY=+7,2‡nCQ_Y = +7,2‡nC.     - Rearranging the formula: r=kQ1Q2Fr = \sqrt{\frac{k Q_1 Q_2}{F}}.     - Answer: r=0,06‡mr = 0,06‡m.
  • \textbf{Example 3: Calculating Charge Magnitude}     - Two identical charges distance 60 mm (0,06‡m0,06‡m), force 9×10−5‡N9 \times 10^{-5}‡N.     - Formula: 9×10−5=(9,0×109)Q2(0,06)29 \times 10^{-5} = \frac{(9,0 \times 10^9) Q^2}{(0,06)^2}.     - Result: Q=6×10−9‡CQ = 6 \times 10^{-9}‡C for each charge.

Multi-Charge Systems and Vectors

  • \textbf{1D Vector Problems}:     - Point charges A, B, and C in a straight line.     - Total force at a point is the vector sum (FNETF_{NET}) of individual forces.     - Must assign a direction as positive (e.g., Left = positive).
  • \textbf{2D Vector Problems}:     - Charges arranged in a right-angled triangle.     - Resultant force (FRESF_{RES}) calculated using: FNET=Fx2+Fy2F_{NET} = \sqrt{F_x^2 + F_y^2}.     - \textbf{Direction}: Calculated using tan⁡(θ)=FyFx\tan(\theta) = \frac{F_y}{F_x}.     - Result must be given as a bearing or compass direction (e.g., "N of E").     - \textbf{Note}: Do not use the geometric distances to calculate the angle of forces; use the force magnitudes.

Advanced Exam Scenario: Equilibrium in a Vertical Plane

  • \textbf{Problem Description}: Sphere Q1Q_1 (+32×10−9‡C+32 \times 10^{-9}‡C) is suspended by a string. Identical sphere Q2Q_2 (−55×10−9‡C-55 \times 10^{-9}‡C) is in a glass tube vertically below it. Distance is 2,5 cm (0,025‡m0,025‡m). Mass of each = 7 g (0,007‡kg0,007‡kg).
  • \textbf{Calculation steps}:     1. \textbf{Electrons removed from Q1Q_1}: n=Qqe=−32×10−9−1,6×10−19=2×1011‡electronsn = \frac{Q}{q_e} = \frac{-32 \times 10^{-9}}{-1,6 \times 10^{-19}} = 2 \times 10^{11}‡\text{electrons}.     2. \textbf{Free-body diagram on Q1Q_1}: Gravity (mgmg) acts down, Tension (TT) acts up, Electrostatic attraction (FEF_E) acts down (toward Q2Q_2).     3. \textbf{Tension Calculation}: FNET=0→T=mg+FEF_{NET} = 0 \rightarrow T = mg + F_E.     4. \textbf{Numerical Solution}: T=(0,007)(9,8)+(9×109)(32×10−9)(55×10−9)(0,025)2=9,39×10−2‡NT = (0,007)(9,8) + \frac{(9 \times 10^9)(32 \times 10^{-9})(55 \times 10^{-9})}{(0,025)^2} = 9,39 \times 10^{-2}‡N.

Practice MCQs and Logic

  • \textbf{Motion of Identical Like Charges}: Two positively charged spheres on a frictionless surface will move away from each other. As distance (rr) increases, FEF_E decreases (F∝1r2F \propto \frac{1}{r^2}). Since F=maF = ma, acceleration decreases. Therefore, they move away with decreasing acceleration.
  • \textbf{Attraction/Repulsion Logic}: Sphere X (negative) attracts Y (therefore Y is positive or neutral) and repels Z (therefore Z is negative). If X attracts Y and repels Z, the conclusion is Y is positive and Z is negative.