34Charge and Force

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CHARGE AND FORCE
▢ Introduction:
❖ Definition: Study of charges at rest = static electricity / electrostatics
❖ Production of Charge: Charges are produced due to actual transfer of electrons

Table 1: Positive and Negative Charge

Charge
Meaning
Mass effect
Positive charge
Deficiency/loss of electrons
Mass slightly decreases
Negative charge
Excess/gain of electrons
Mass slightly increases
❖ Important Point: Mass of negatively charged body is slightly greater than mass of positively charged body
▢ Properties of Charges:
❖ Quantization of Charge:
◉ Formula: \(q=\pm ne\)
◉ Meaning: \(n\) is an integer
◉ Possible Charges: \(q=\pm1e,\pm2e,\pm3e,...\)
◉ Elementary Charge: \(e=1.6\times10^{-19}\ C\)
◉ Important Point: Charge less than one elementary charge is not possible on a body
❖ Conservation of Charge:
◉ Statement: Total electric charge of an isolated system always remains constant
◉ Atomic Neutrality: Every atom is electrically neutral because number of electrons = number of protons
◉ Rubbing Example: When glass rod is rubbed with silk, positive charge on glass = negative charge on silk
◉ Conclusion: Charges are produced in equal and unlike pairs
❖ Interaction:

Table 1: Interaction of Charges

Charges
Force
Like charges
Repel
Unlike charges
Attract
Sure test of electrification
Repulsion
❖ Invariance of Charge: Magnitude of charge is not affected by motion, unlike mass
❖ Field Produced by Charge:

Table 1: Charge Motion and Fields

Condition of charge
Produces
Charge at rest
Electric field only
Charge in uniform/unaccelerated motion
Electric field + magnetic field
Accelerated charge
Electric field + magnetic field + electromagnetic radiation
▢ Charging Preference:
❖ Rule: Higher substance in series becomes positive; lower substance becomes negative
❖ Series:
    ◈ _*type: number
  1. Fur
  2. Flannel
  3. Glass
  4. Cotton
  5. Paper
  6. Silk
  7. Man
  8. Wood
  9. Metal
  10. Rubber
  11. Resin
  12. Amber
  13. Sulphur
  14. Ebonite
▢ Charging of Body:
❖ **table:
    ❖ Induction:
    ◉ Preference: Charging by induction is preferable because same charged body can charge many bodies without loss of charge
    ◉ Formula: \(q*{induced}=-q*{inducing}\left(1-\frac{1}{\epsilon_r}\right)\)
    ◉ **table:
      ◉ Important Points:
      • •
        Negative sign indicates induced charge is opposite in nature
      • •
        In induction process, net induced charge on body is zero
      • •
        In induction process, charge and mass of inducing body remain same
      • •
        Potential of charged body decreases during induction
      • •
        Electrical induction = temporary electrification under influence of charged body
      ▢ Charge Density:
      ❖ **table:
        ▢ Coulomb's Law:
        ❖ Statement: Force between two point charges is directly proportional to product of charges and inversely proportional to square of distance between them
        ❖ Proportionality: \(F\propto\frac{q_1q_2}{r^2}\)
        ❖ **table:
          ◈ caption: Constants
          ◈ data:
            1. Quantity
            2. Value / Meaning
            1. \(\frac{1}{4\pi\epsilon_0}\)
            2. \(9\times10^9\ Nm^2C^{-2}\)
            1. \(\epsilon_0\)
            2. \(8.85\times10^{-12}\ C^2N^{-1}m^{-2}\)
            1. \(\epsilon_0\)
            2. Permittivity of free space
            1. \(\epsilon_r\)
            2. Relative permittivity / dielectric constant
            1. \(\epsilon_r\)
            2. No unit and no dimension
        ❖ Important Points:
        • •
          Coulomb's law is valid only for point charges
        • •
          Coulomb's law is valid for distance greater than \(10^{-15}\ m\)
        • •
          Coulomb's force does not depend on mass of charges
        • •
          Coulomb's force between two charges does not depend on presence or absence of other charges
        • •
          For two charges, forces are equal and opposite: \(F_1=-F_2\)
        • •
          Coulomb force may be attractive or repulsive
        • •
          Coulomb force depends on medium between charges
        ▢ Charge Units:

        Table 1: Important Unit Relations

        Quantity
        Relation
        1 coulomb
        \(6.25\times10^{18}\) electrons
        CGS unit of charge
        esu / stat coulomb
        1 coulomb
        \(3\times10^9\ esu\)
        1 volt
        \(\frac{1}{300}\ statvolt\)
        1 farad
        \(9\times10^{11}\ statfarad\)
        ▢ Conductors and Earthing:
        ❖ Charge on Conductor: Charge resides only on outer surface of charged conductor
        ❖ Positive Conductor Earthed: Electrons flow from earth to conductor
        ❖ Identical Conductors Sharing Charge:

        Table 1: Charge Sharing

        Initial charges
        Final charge on each conductor
        \(q_1\) and \(q_2\)
        \(\frac{q_1+q_2}{2}\)
        \(q_1\) and \(-q_2\)
        \(\frac{q_1-q_2}{2}\)
        ▢ Dielectrics:

        Table 1: Types of Dielectrics

        Type
        Examples
        Description
        Non-polar dielectrics
        \(N_2,O_2\), benzene, methane
        Centre of positive charge coincides with centre of negative charge
        Polar dielectrics
        \(H_2O,CO_2,NH_3,HCl\)
        Centre of positive charge does not coincide with centre of negative charge
        ❖ Important Points:
        • •
          Non-polar dielectric can be polarized by external electric field
        • •
          Dielectric constant for conductor = \(\infty\)
        • •
          Dielectric constant for insulator > 1
        • •
          Coulomb force is maximum in air/vacuum
        • •
          In dielectric medium, Coulomb force decreases
        ▢ Force Comparison:
        ❖ _*table:
          ❖ Important Points:
          • •
            Inside nucleus, nuclear force and electrostatic force both exist
          • •
            Gravitational force is neglected for proton, electron and neutron compared to Coulomb and nuclear forces
          ▢ Special Cases in Coulomb Force:

          Table 1: Useful Results

          Condition
          Result
          One charge very large compared to another
          Even like charges may attract due to induction
          Charge \(Q\) divided into two parts for maximum force
          Each part = \(\frac{Q}{2}\)
          Two small spheres with constant charge: force in air : force in medium
          \(K:1\)
          Two small spheres at constant potential: force in air : force in medium
          \(1:K\)
          ▢ Pith Ball Problems:

          Table 1: Two Identical Charged Pith Balls

          Condition
          Formula / Result
          Two balls each mass \(m\), charge \(q\), thread length \(l\)
          \(\frac{F}{mg}=\tan\theta\approx\sin\theta=\frac{x}{2l}\)
          In gravity-free space
          Angle between strings = \(180^\circ\)
          Tension in each string in gravity-free space
          \(T=\frac{1}{4\pi\epsilon_0}\frac{q^2}{4l^2}\)
          In liquid, same distance remains
          \(K=\frac{\sigma}{\sigma-\rho}\)
          ❖ Symbols:
          • •
            \(x\) = distance between balls
          • •
            \(\sigma\) = density of body
          • •
            \(\rho\) = density of liquid
          • •
            \(K\) = dielectric constant
          ▢ Equilibrium of Charges:

          Table 1: Charge Equilibrium Configurations

          Configuration
          Charge required for equilibrium
          Three identical charges \(+q\) at corners of equilateral triangle; charge at centroid
          \(Q=-\frac{q}{\sqrt3}\)
          If \(Q=-q\) at centroid
          All three corner charges move to centroid
          Four identical charges \(+q\) at corners of square; charge at centre
          \(Q=-\frac{q}{4}(1+2\sqrt2)\)
          Charge \(q\) at centre of line joining two equal charges \(Q\)
          \(q=-\frac{Q}{4}\)
          ❖ Trick:
          • •
            If end charges are positive, centre charge must be negative
          • •
            If end charges are negative, centre charge must be positive
          • •
            Magnitude of centre charge = one-fourth of end charge in equal-charge line problem
          ▢ Zero Force Point:
          ❖ **table:
            ❖ Trick Formula:
            ◉ Formula: \(x=\frac{d}{1\pm\sqrt{\frac{q*{lower}}{q*{higher}}}}\)
            ◉ Meaning: \(x\) = distance from higher charge
            ◉ Sign:
            • •
              Use + for like charges
            • •
              Use − for unlike charges
            ▢ Dielectric Slab Between Charges:
            ❖ Single Slab: If dielectric slab of thickness \(t\) and dielectric constant \(K\) is placed between charges, effective separation changes due to dielectric
            ❖ Multiple Slabs: For multiple slabs, effective distance is calculated by adding individual apparent/effective thickness contributions
            ❖ Important Point: To keep same force in vacuum and medium of dielectric constant \(K\), separation must be adjusted because \(F\propto\frac{1}{Kr^2}\)
            ▢ Gold Leaf Electroscope:
            ❖ Use: Divergence of gold leaf electroscope is used to study charge

            Table 1: Gold Leaf Electroscope Test

            Observation
            Conclusion
            Leaves do not diverge
            Body is uncharged
            Leaves diverge
            Body is charged
            ▢ Soap Bubble and Charge:
            ❖ Observation: If soap bubble is given small positive or negative charge, its radius increases
            ❖ Reason: Repulsion between charges expands bubble
            ❖ Surface Tension: Surface tension decreases
            ❖ Explanation: Bubble expands to minimize surface tension while balancing outward electrostatic pressure
            ▢ Faraday Ice-Pail Experiment:
            ❖ Purpose: Establishes relation between inducing and induced charge
            ❖ Results:
            • •
              Induced positive charge = induced negative charge
            • •
              Induced charge = inducing charge
            ▢ Van de Graaff Generator:
            ❖ Definition: Powerful machine used for generating very high positive potential
            ❖ Potential: \(10^6\ volts\)
            ▢ Everyday Static Electricity:
            ❖ _*table:
              ▢ Read and Digest:
              ❖ **table:
                ▢ High-Yield Recall:
                ❖ **table:
                  Q1.
                  A positively charged particle moving east enters a region of uniform magnetic field directed vertically upwards. The particle will:
                  📅IOM 2013
                  Q2.
                  A moving charged particle experiences force in:
                  📅IOM 2012
                  Q3.
                  Two ebonite rods A and B of radii 1.5cm and 2cm rubbed together will:
                  📅IOM 2010
                  Q4.
                  A thin water stream from tap is attracted by a rod placed near it. The rod is:
                  📅IOM 2010
                  Q5.
                  Neutral metal gains 32μC charge when rubbed with wool. Electron transfer count:
                  📅MOE 2068
                  Q6.
                  Force between electron and proton at 0.53Å separation:
                  📅MOE 2068
                  Q7.
                  Ratio of electric to gravitational force between electrons:
                  📅MOE 2010
                  Q8.
                  Force between two 1μC charges at 1m separation:
                  📅MOE 2009
                  Q9.
                  Positive charge Q released between two fixed -q charges at (0,a) and (0,-a): Two equal negative charged —q are fixed at the points (0, a) & (0, —a) on the y-axis. A positive charge Q is released from rest at the point (2a, 0) on x-axis. The charge Q will
                  📅BP 2010
                  Q10.
                  Glass rod becomes positive when rubbed with silk due to:
                  📅BP 2010
                  Q11.
                  Work done moving charge on equipotential surface:
                  📅KU 2011
                  Q12.
                  Specific charge of electron:
                  📅BP 2013
                  Q13.
                  Charged sphere (radius a, charge Q) contacts uncharged sphere (radius b). Remaining charge on A:
                  📅IE 2011
                  Q14.
                  Distance between plates with E=106 V/m and ΔV=103 V:
                  📅KU 2010
                  Q15.
                  Equipotential surface for point charge at large distance appears:
                  📅IE-05
                  Q16.
                  If the potentil of metallic sphere 100V with respect to far away. Calculate the surface charge density if the radius of sphere 2cm.
                  📅IE-06
                  Q17.
                  Identical charges are kept at each corner of a cube of length a. The net potential at the center of cube is
                  📅
                  Q18.
                  Correct statement about matter waves:
                  📅BPKIHS-94
                  Q19.
                  On penetrating a uniformly charged sphere, the electrical field strength E
                  📅BPKIHS-94
                  Q20.
                  Five balls numbered 1 to 5 are suspended using separate threads. Pairs (1,2), (2, 4), (4,1) show electrostatic attraction, while pairs (2,3) and (4,5) show repulsion. Therefore, ball 1 must be
                  📅BPKIHS-07
                  Q21.
                  Characteristic of electric field lines:
                  📅IE-02
                  Q22.
                  Two concentric spheres are carrying a charge Q and q. When the two spheres are joined by a wire, the charge on the outer surface of the inner sphere is
                  📅IE-03
                  Q23.
                  When two bodies are rubbed against each other they are electrified. Which one of the followin is correct?
                  📅IOM 2003
                  Q24.
                  Divergence of gold leaf is used to study
                  📅IOM 2002
                  Q25.
                  When a woolen sweater worm over a nylon shirt is removed, sparking is observed due to
                  📅IOM 1999
                  Q26.
                  A charge q is placed at the centre of the line joining two equal charges Q. The system of three charges will be in equilibrium if q is equal to:
                  📅MOE
                  Q27.
                  Where a charge Q is kept between +q and +nq (n > 0) so that net force experienced by +Q charge will be zero. Assume distance between charges is r
                  📅
                  Q28.
                  Force between two identical spheres carrying charge +10μC and -20μC, kept at certain distance is F. If these two spheres are made in contact and then separated to the distance, force is F'. The ratio of these two force is
                  📅
                  Q29.
                  A charge Q = 120μC is to be divided into two parts q and Q—q. These parts are kept at certain distance then for what value Of q the force between them will be maximum?
                  📅
                  Q30.
                  The distance between two. charges is increased by 25% then the percentage change in force is
                  📅
                  Q31.
                  Coloumb force between two charges kept at certain distance is F. If 'half of separåtion between them is filled with dielectric of constant εr = 4, then the new Coloumb's force is
                  📅
                  Q32.
                  There are two charges +1μC and +3μC. The ratio of the forces acting on them will be
                  📅
                  Q33.
                  A charge q1 exerts some force on a second charge q2. If third charge q3 is brought near, the force q1 exerted on q2:
                  📅
                  Q34.
                  Two equally charged identical metal spheres A and B repel each other with a force 3×10-5 N. Another identical unchanged sphere C is touched with A and then placed at the mid-point between A and B. Net force on C is(initial force ):
                  📅
                  Q35.
                  A polythene piece, rubbed with wool is found to have negative charge of 4×10-7C. The number of electrons transferred from wool to polythene is
                  📅
                  Q36.
                  Two insulated metal spheres of radii 10cm and 15cm charged .to a potential of 150V and respectively are connected by means of a metallic wire. What is ihe charge on the first sphere?
                  📅
                  Q37.
                  Two identical Α and Β are supported on insulating stands and placed in contact. What kind of charges will Α and Β develop when a negatively charged ebonite rod is brought near Α?
                  📅
                  Q38.
                  Three charges each of are Placed at the corners of an equilateral triangle. lf the force between any two charges be F, then the net force on either charge will be
                  📅
                  Q39.
                  Two equally charged identical metal sphere Α and Β rcpel each other witll a force F. Another identical unchanged sphcre C is touched with Α and then with Β and finally removed. What is the new force of repulsion betwccn Α and Β?
                  📅
                  Q40.
                  A positively charged ball hangs from a silk thread. We put a +ve test charge qo at a then it can be point and measure F/qo then it can be predicted that the electric field strength E
                  📅IE
                  Q41.
                  Two point charges placed at n. certain distance 'r' in air exert a force of F on each other. Then the distance at which these charge will experience the same force in a medium of dielectric constant K is
                  📅
                  Q42.
                  A charged liarticle q is shot towards another fixed charge particle Q with a speed 'v'. It approaches Q upto a closest distance 'r' and then returns. If q were given a speed 2v, the closest distance of approach would be
                  📅
                  Q43.
                  The ratio of the forces between two small spheres charged to constant potentials in air and in medium of dielectric constant K is:
                  📅
                  Q44.
                  The fig. shows the variation of potential V with separation r of two molecules.
                  📅KU 2015
                  Q45.
                  Each of two small non conducting spheres is charged positively, the combined charge being 40 PC. When the two spheres are 50cm apart, each sphere is repelled from the other by a force of magnitude 2.0N. The magnitude of the smaller of the two charges is
                  📅KU 2015