Electric Charges and Fields Class 12 Notes

• $\text{Scalar and additive:}$
• charge is a scalar; the net charge of a system is the algebraic sum of individual charges
• $q_{net}=\sum q_i$
• .
• $\text{Quantised:}$
• charge always exists in integer multiples of the elementary charge,
• $q=\pm ne$
• , with
• $e=1.6\times 10^{-19}\text{C}$
• .
• $\text{Conserved:}$
• the total charge of an isolated system is constant; charge can be transferred but never created or destroyed.
• $\text{Invariant:}$
• charge does not depend on the speed of the charged body.
• $\text{Always with mass:}$
• charge cannot exist without mass, though mass can exist without charge.

MCQ Questions

1 Mark Questions

1. 1
   State the SI unit of electric charge.
2. 2
   How does a glass rod become positively charged when rubbed with silk?
3. 3
   What is the magnitude of charge on an electron?
4. 4
   Why is repulsion taken as a sure test of electrification but attraction is not?

2 Mark Questions

1. 1
   State and explain three basic properties of electric charge.
2. 2
   Why is charge quantised? Explain with q = ne.
3. 3
   Does the mass of a positively charged body differ from its neutral mass? Explain.
4. 4
   Distinguish between conductors and insulators with reference to charge flow.

3 Mark Questions

1. 1
   How many electrons must be given to a neutral body so that it acquires a charge of 4.0 pC?
2. 2
   Explain, with examples, that charge is an additive, conserved and quantised physical quantity.
3. 3
   A body has an excess of 1.5 × 10⁷ electrons. Find the charge on the body.

Here $k=9\times 10^9{\text{N m}}^2{\text{C}}^{-2}$ and ${\epsilon }_0=8.85\times 10^{-12}{\text{C}}^2{\text{N}}^{-1}{\text{m}}^{-2}$ is the permittivity of free space.${\vec{F}}_{12}$ is the force on charge 1 due to charge 2, directed along the vector from 2 to 1.

MCQ Questions

1 Mark Questions

1. 1
   State Coulomb's law in electrostatics.
2. 2
   Define relative permittivity (dielectric constant) of a medium.
3. 3
   Write the value of the Coulomb constant k and of ε₀.
4. 4
   What does the principle of superposition of forces state?

2 Mark Questions

1. 1
   Two identical balls with equal mass and charge are suspended by insulating strings of equal length. State how the electrostatic force and weight combine in equilibrium (Lami's theorem approach).
2. 2
   Compare the electrostatic and gravitational forces between two protons.
3. 3
   A charge Q is to be divided into two parts so that the force between them is maximum. Find the division.
4. 4
   Why does the force between two charges decrease when an insulating medium replaces vacuum?

3 Mark Questions

1. 1
   Calculate the Coulomb force between two α-particles (q = +3.2 × 10⁻¹⁹ C each) separated by 3.2 × 10⁻¹⁵ m.
2. 2
   Three charges +Q, −Q and +q are placed at the vertices of an equilateral triangle. Using the principle of superposition, find the net force on +q.
3. 3
   Two identical spheres P and Q each carry charge q. A third identical uncharged sphere R is touched first with P and then with Q. Find the final force between P and Q in terms of the initial force.

For a positive source charge the field points radially away; for a negative source it points radially towards the charge. A force $\vec{F}=q\vec{E}$ acts on a charge q placed in the field, along the field for +q and opposite to it for −q.

MCQ Questions

1 Mark Questions

1. 1
   Define electric field intensity at a point.
2. 2
   Write the expression for the electric field due to a point charge.
3. 3
   Give the SI unit of electric field.
4. 4
   What is the direction of the force on a negative charge placed in an electric field?

2 Mark Questions

1. 1
   Define electric field as the force per unit positive test charge and justify why the test charge must be infinitesimally small.
2. 2
   Two point charges q₁ and q₂ are placed at two corners of a square. Using superposition of fields, express the field at the empty corner.
3. 3
   A uniform electric field E is set up between two parallel plates. An electron enters symmetrically with speed v₀ and the plates have length l. Derive the angle of deviation of the electron using m, e and E.

3 Mark Questions

1. 1
   Derive the expression E = q/(4πε₀r²) for the field of a point charge from Coulomb's law.
2. 2
   An electron of charge e and mass m enters a uniform electric field E between two parallel plates of length l with speed v₀. Show that it emerges deviated by an angle θ = tan⁻¹(eEl/mv₀²).
3. 3
   Calculate the electric field at the midpoint of the line joining two charges +q and −q separated by a distance d.

For a continuous distribution, the field is the vector integral over all elements dq: $\vec{E}=\frac{1}{4\pi {\epsilon }_0}\int \frac{dq}{r^2}\hat{r}$, where $dq=\lambda dl=\sigma dA=\rho dV$ depending on the geometry.

• $x=0$
• (centre of the ring): E = 0, by symmetry.
• $x\gg R$
• :
• $E\approx \frac{1}{4\pi {\epsilon }_0}\frac{q}{x^2}$
• — the ring behaves like a point charge.
• E is maximum at
• $x=R/\sqrt{2}$
• with
• $E_{max}=\frac{2}{3\sqrt{3}}\frac{q}{4\pi {\epsilon }_0{R}^2}$
• .

• $x\gg a$
• :
• $E\approx \frac{1}{4\pi {\epsilon }_0}\frac{q}{x^2}$
• — far away the line behaves like a point charge.
• Infinite line
• $(a\to \infty )$
• :
• $E=\frac{\lambda }{2\pi {\epsilon }_{0}x}$
• — falls off as 1/x.

MCQ Questions

1 Mark Questions

1. 1
   Define linear, surface and volume charge densities and give their SI units.
2. 2
   Write the expression for the electric field on the axis of a ring of charge.
3. 3
   At what distance on the axis of a charged ring is the field maximum?
4. 4
   Write the field of an infinitely long line charge.

2 Mark Questions

1. 1
   Derive E = qx/[4πε₀(x² + R²)^{3/2}] for the electric field on the axis of a uniformly charged ring.
2. 2
   Show that for x ≫ R the ring field becomes that of a point charge.
3. 3
   Derive the electric field on the perpendicular bisector of a uniformly charged line of length 2a.

3 Mark Questions

1. 1
   A charge Q is uniformly distributed on a circular ring of radius a. Derive the electric field at a point on its axis at distance x from the centre and hence obtain the value at the centre.
2. 2
   Obtain the electric field at a distance r from an infinitely long uniformly charged line of linear charge density λ using proper integration, and show it behaves like a point charge for r ≫ a.
3. 3
   A non-conducting rod of length L carries uniform positive charge density λ and total charge Q, lying along the x-axis. A point P is at distance x₀ from one end on the axis. Using superposition, express the magnitude of the electric field at P.

• Field lines start at positive charges and end on negative charges; they extend from a lone positive charge to infinity and from infinity to a lone negative charge.
• No two field lines intersect — the field has a unique direction at every point.
• The tangent to a field line at a point gives the direction of the field.
• Line density indicates field strength: crowded lines mean a stronger field.
• Near a conductor field lines are perpendicular to the surface and do not enter the conductor.
• The number of lines leaving (or entering) a charge is proportional to the magnitude of the charge.
• Static-charge field lines never form closed loops and do not enter conductors.

MCQ Questions

1 Mark Questions

1. 1
   What is an electric line of force?
2. 2
   Why do two electric field lines never intersect?
3. 3
   Where do electric field lines start and end?
4. 4
   State how the number of field lines is related to the magnitude of charge.

2 Mark Questions

1. 1
   Sketch the field lines of (a) a single positive charge, (b) a single negative charge, and (c) an electric dipole.
2. 2
   Why is the line density of the field a measure of the field strength?
3. 3
   Why do static field lines never form closed loops?

3 Mark Questions

1. 1
   Two point charges q₁ and q₂ are shown with 6 field lines originating from q₁ and 18 lines ending on q₂. Deduce the signs and the ratio q₁/q₂.
2. 2
   Describe the electric field line pattern for a system of two equal positive charges and for a dipole, and explain why the lines never cross.
3. 3
   Explain with the help of field lines why the electric field is zero inside a closed conducting surface.

Flux is positive where field lines leave the surface and negative where they enter. For a closed surface the normal is chosen outward. The area element dA on a sphere subtends a solid angle $\Delta \Omega =\Delta A/r^2$, and the total solid angle around a point is 4π sr.

MCQ Questions

1 Mark Questions

1. 1
   Define electric flux through a surface.
2. 2
   Write the unit and dimensional formula of electric flux.
3. 3
   What is the flux through a surface placed parallel to a uniform electric field?
4. 4
   What is a solid angle? What is its unit?

2 Mark Questions

1. 1
   Derive the expression Φ_E = EA cosθ for the flux through a tilted surface in a uniform field.
2. 2
   A non-uniform field E = 3.0x i + 4.0 j pierces the faces of a cube. Find the flux through (a) the right face at x = 3.0 m of area 4.0 m² and (b) the top face.
3. 3
   Explain why the flux through a closed surface is positive for outward lines and negative for inward lines.

3 Mark Questions

1. 1
   Consider the non-uniform field E = 3.0x i + 4.0 j newton per coulomb (x in metres) piercing the faces of a cube with faces at x = 1.0 m, x = 3.0 m and area 4.0 m². Calculate the net flux through the right, left and top faces and the net charge enclosed.
2. 2
   Show that the flux of a point charge q through a spherical surface of radius r enclosing it is q/ε₀, independent of r.
3. 3
   A small area ΔA on a sphere of radius r₁ subtends a solid angle ΔΩ at the centre. Express ΔA in terms of r₁ and ΔΩ, and find the total solid angle of a full sphere.

For a point charge Q at the centre of a sphere, E = Q/4πε₀r² on the surface and the flux is ${\Phi }_E=E(4\pi r^2)=Q/{\epsilon }_0$ — independent of r. This is why Gauss's law is exact and powerful for symmetric distributions.

MCQ Questions

1 Mark Questions

1. 1
   State Gauss's law of electrostatics.
2. 2
   Does the flux through a Gaussian surface depend on charges outside it?
3. 3
   Write Gauss's law for a point charge at the centre of a sphere.
4. 4
   Why is the flux through a closed surface independent of the distribution of the enclosed charge?

2 Mark Questions

1. 1
   Prove that the flux of a point charge Q through a spherical Gaussian surface is Q/ε₀.
2. 2
   Five charged lumps lie near a Gaussian surface: q₁ = q₄ = +3.1 nC, q₂ = q₅ = −5.9 nC, q₃ = −3.1 nC; q₃, q₁ and a neutral coin are inside. Compute the net flux.
3. 3
   Explain how the shape of a Gaussian surface affects the total flux (or does not) and why.

3 Mark Questions

1. 1
   Using Gauss's law, show that the net flux through a closed surface enclosing charges q₁ = +3.1 nC, q₂ = −5.9 nC and q₃ = −3.1 nC is −670 N·m²/C (ε₀ = 8.85 × 10⁻¹²).
2. 2
   State the conditions under which Gauss's law may be used to easily find the electric field, and apply it to a uniformly charged thin spherical shell.
3. 3
   A point charge is placed at the centre of a cube. Find the flux through one face in terms of the total flux.

• $\text{Infinite non-conducting sheet:}$
• $E=\frac{\sigma }{2{\epsilon }_0}$
• , uniform and independent of distance, on both sides of the sheet.
• $\text{Infinite conducting sheet:}$
• $E=\frac{\sigma }{{\epsilon }_0}$
• outside the surface.
• $\text{Infinite line charge:}$
• $E=\frac{\lambda }{2\pi {\epsilon }_{0}r}$
• radially outward, varying as 1/r.
• $\text{Spherical shell (r < R):}$
• E = 0 inside; outside the shell
• $E=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}$
• .
• $\text{Solid non-conducting sphere (r ≤ R):}$
• $E=\frac{Qr}{4\pi {\epsilon }_0{R}^3}=\frac{\rho r}{3{\epsilon }_0}$
• — field grows linearly with r inside.
• $\text{Outside the solid sphere (r ≥ R):}$
• $E=\frac{1}{4\pi {\epsilon }_0}\frac{Q}{r^2}$
• .

MCQ Questions

1 Mark Questions

1. 1
   Write the electric field due to an infinite non-conducting sheet of charge density σ.
2. 2
   What is the electric field inside a uniformly charged spherical shell?
3. 3
   Write the field inside a uniformly charged non-conducting solid sphere at distance r from the centre.
4. 4
   State the field of an infinite line charge at distance r.

2 Mark Questions

1. 1
   Using Gauss's law, derive E = σ/2ε₀ for an infinite non-conducting sheet using a cylindrical Gaussian surface (pillbox).
2. 2
   Derive the electric field inside and outside a uniformly charged thin spherical shell by Gauss's law.
3. 3
   Derive E = λ/2πε₀r for an infinitely long line charge using a coaxial cylindrical Gaussian surface.

3 Mark Questions

1. 1
   Derive the electric field at a point inside and outside a uniformly charged non-conducting solid sphere of radius R and total charge Q by Gauss's law, and sketch E versus r.
2. 2
   Use Gauss's law to find the field of an infinitely long rod of uniform linear charge density λ at a distance r, and verify it agrees with E = λ/2πε₀r.
3. 3
   A thin spherical shell of radius a carries charge +Q uniformly on its surface. Obtain E at (i) r < a and (ii) r ≥ a by Gauss's law and comment on the discontinuity at r = a.

Exercises (NCERT Q1.1–Q1.16)

• 1.1) What is the force between two small charged spheres having charges of 2 × 10⁻⁷ C and 3 × 10⁻⁷ C placed 30 cm apart in air?
• 1.2) The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge −0.8 µC in air is 0.2 N. (a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first?
• 1.3) Check that the ratio ke²/Gmₑmₚ is dimensionless. Look up a table of physical constants and determine the value of this ratio. What does the ratio signify?
• 1.4) (a) Explain the meaning of the statement 'electric charge of a body is quantised'. (b) Why can one ignore quantisation of electric charge when dealing with macroscopic, i.e. large-scale charges?
• 1.5) When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge.
• 1.6) Four point charges q_A = 2 µC, q_B = −5 µC, q_C = 2 µC and q_D = −5 µC are located at the corners of a square ABCD of side 10 cm. What is the force on a charge of 1 µC placed at the centre of the square?
• 1.7) (a) An electrostatic field line is a continuous curve: a field line cannot have sudden breaks. Why? (b) Explain why two field lines never cross each other at any point.
• 1.8) Two point charges q_A = 3 µC and q_B = −3 µC are located 20 cm apart in vacuum. (a) What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude 1.5 × 10⁻⁹ C is placed at this point, what is the force experienced by the test charge?
• 1.9) A system has two charges q_A = 2.5 × 10⁻⁷ C and q_B = −2.5 × 10⁻⁷ C located at points A (0, 0, −15 cm) and B (0, 0, +15 cm), respectively. What are the total charge and electric dipole moment of the system?
• 1.10) An electric dipole with dipole moment 4 × 10⁻⁹ C m is aligned at 30° with the direction of a uniform electric field of magnitude 5 × 10⁴ N/C. Calculate the magnitude of the torque acting on the dipole.
• 1.11) A polythene piece rubbed with wool is found to have a negative charge of 3 × 10⁻⁷ C. (a) Estimate the number of electrons transferred (from which to which?). (b) Is there a transfer of mass from wool to polythene?
• 1.12) Two insulated charged copper spheres A and B have their centres separated by a distance of 50 cm. (a) What is the mutual force of electrostatic repulsion if the charge on each is 6.5 × 10⁻⁷ C? (b) What is the force of repulsion if each sphere is charged double the above amount and the distance between them is halved?
• 1.13) Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B?
• 1.14) The figure below shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge-to-mass ratio?
• 1.15) Consider a uniform electric field E = 3 × 10³ î N/C. (a) What is the flux of this field through a square of 10 cm on a side whose plane is parallel to the yz-plane? (b) What is the flux through the same square if the normal to its plane makes a 60° angle with the x-axis?
• 1.16) What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes?

Additional Exercises (NCERT Q1.17–Q1.34)

• 1.17) Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 10³ N·m²/C. (a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or why not?
• 1.18) A point charge +10 µC is at a distance 5 cm directly above the centre of a square of side 10 cm, as shown in the figure. What is the magnitude of the electric flux through the square? (Hint: think of the square as one face of a cube with edge 10 cm.)
• 1.19) A point charge of 2.0 µC is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface?
• 1.20) A point charge causes an electric flux of −1.0 × 10³ N·m²/C to pass through a spherical Gaussian surface of 10.0 cm radius centred on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge?
• 1.21) A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the centre of the sphere is 1.5 × 10³ N/C and points radially inward, what is the net charge on the sphere?
• 1.22) A uniformly charged conducting sphere of 2.4 m diameter has a surface charge density of 80.0 µC/m². (a) Find the charge on the sphere. (b) What is the total electric flux leaving the surface of the sphere?
• 1.23) An infinite line charge produces a field of 9 × 10⁴ N/C at a distance of 2 cm. Calculate the linear charge density.
• 1.24) Two large, thin metal plates are parallel and close to each other. On their inner faces the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10⁻²² C/m². What is E in (a) the outer region of the first plate, (b) the outer region of the second plate, and (c) between the plates?
• 1.25) An oil drop of 12 excess electrons is held stationary under a constant electric field of 2.55 × 10⁴ N/C (Millikan's oil-drop experiment). The density of the oil is 1.26 g/cm³. Estimate the radius of the drop. (g = 9.81 m/s²; e = 1.60 × 10⁻¹⁹ C.)
• 1.26) Which among the curves shown in the figure cannot possibly represent electrostatic field lines? Give reasons for your answers.
• 1.27) In a certain region of space the electric field is along the z-direction throughout. The magnitude of E is not constant but increases uniformly along the positive z-direction at the rate of 10⁵ N/C per metre. What are the force and torque experienced by a system having a total dipole moment equal to 10⁻⁷ C m in the negative z-direction?
• 1.28) (a) A conductor A with a cavity is given a charge Q. Show that the entire charge must appear on the outer surface of the conductor. (b) Another conductor B with charge q is inserted into the cavity keeping B insulated from A. Show that the total charge on the outside surface of A is Q + q. (c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way.
• 1.29) A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (σ/2ε₀) n̂, where n̂ is the unit vector in the outward normal direction and σ is the surface charge density near the hole.
• 1.30) Obtain the formula for the electric field due to a long thin wire of uniform linear charge density λ without using Gauss's law. [Hint: use Coulomb's law directly and evaluate the necessary integral.]
• 1.31) A proton and a neutron are made of 'up' quarks (u, charge +2/3 e) and 'down' quarks (d, charge −1/3 e). Suggest a possible quark composition of a proton and of a neutron.
• 1.32) (a) Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (E = 0) of the configuration. Show that the equilibrium of the test charge is necessarily unstable. (b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.
• 1.33) A particle of mass m and charge −q enters the region between two charged plates initially moving along the x-axis with speed vₓ. The length of each plate is L and a uniform electric field E is maintained between the plates. Show that the vertical deflection of the particle at the far edge of the plate is qEL²/(2mvₓ²). Compare this motion with the motion of a projectile in a gravitational field.
• 1.34) Suppose the particle in Exercise 1.33 is an electron projected with velocity vₓ = 2.0 × 10⁶ m/s. If E between the plates separated by 0.5 cm is 9.1 × 10² N/C, where will the electron strike the upper plate? (|e| = 1.6 × 10⁻¹⁹ C, mₑ = 9.1 × 10⁻³¹ kg.)

Answers at a glance

1. 1
   Q1.1 — F = 6 × 10⁻³ N (repulsive).
2. 2
   Q1.2 — (a) r = 0.12 m; (b) 0.2 N, equal and opposite (Newton's third law).
3. 3
   Q1.3 — Dimensionless; ratio ≈ 2.3 × 10³⁹ (electric to gravitational force between a proton and an electron).
4. 4
   Q1.4 — (a) Charge exists only as integral multiples of e; (b) macroscopic charges are huge compared with e, so the granularity is unobservable.
5. 5
   Q1.5 — Rubbing only transfers electrons; equal positive and negative charges appear, so the total charge is conserved.
6. 6
   Q1.6 — Net force = 0 N (the forces from the four corner charges cancel by symmetry).
7. 7
   Q1.7 — (a) A charge moves continuously, so the field line cannot break; (b) crossing would give two directions of E at one point.
8. 8
   Q1.8 — (a) E = 5.4 × 10⁶ N/C along OB; (b) F = 8.1 × 10⁻³ N along OA.
9. 9
   Q1.9 — Total charge = 0; p = 7.5 × 10⁻⁸ C m along the +z-axis.
10. 10
    Q1.10 — τ = pE sinθ = 10⁻⁴ N·m.
11. 11
    Q1.11 — (a) n ≈ 1.87 × 10¹² electrons, transferred from wool to polythene; (b) yes, mass transferred ≈ 1.7 × 10⁻¹⁸ kg (negligible).
12. 12
    Q1.12 — (a) F = 1.52 × 10⁻² N; (b) F = 0.243 N (16 times larger).
13. 13
    Q1.13 — F ≈ 5.7 × 10⁻³ N (3/8 of the initial force; A carries q/2, B carries 3q/4).
14. 14
    Q1.14 — Particles 1 and 2 are negative, particle 3 is positive; particle 3 has the highest charge-to-mass ratio.
15. 15
    Q1.15 — (a) Φ = 30 N·m²/C; (b) Φ = 15 N·m²/C.
16. 16
    Q1.16 — Net flux = 0 (uniform field: lines entering equal lines leaving).
17. 17
    Q1.17 — (a) q = 0.07 µC; (b) no — zero net flux only means zero net enclosed charge.
18. 18
    Q1.18 — Φ = 1.88 × 10⁵ N·m²/C.
19. 19
    Q1.19 — Φ = 2.26 × 10⁵ N·m²/C.
20. 20
    Q1.20 — (a) Same, −10³ N·m²/C (flux is independent of the Gaussian radius); (b) q = −8.85 nC.
21. 21
    Q1.21 — q = 6.67 × 10⁻⁹ C = 6.67 nC (negative, radially inward).
22. 22
    Q1.22 — (a) Q = 1.45 × 10⁻³ C; (b) Φ = 1.63 × 10⁸ N·m²/C.
23. 23
    Q1.23 — λ = 10 µC/m.
24. 24
    Q1.24 — E = 0 in both outer regions; between the plates E = σ/ε₀ = 1.92 × 10⁻¹⁰ N/C.
25. 25
    Q1.25 — r ≈ 9.82 × 10⁻⁷ m.
26. 26
    Q1.26 — Only (c) is valid; (a), (b), (d) and (e) cannot represent electrostatic field lines.
27. 27
    Q1.27 — F = −10⁻² N (opposite to z); τ = 0.
28. 28
    Q1.28 — (a) E = 0 inside the conductor compels q_enc = 0 on the inner surface; (b) outer surface charge = Q + q; (c) shield the instrument by enclosing it in a closed metal box (Faraday cage).
29. 29
    Q1.29 — E_hole = σ/2ε₀ along n̂.
30. 30
    Q1.30 — E = λ/2πε₀r.
31. 31
    Q1.31 — Proton = uud; neutron = udd.
32. 32
    Q1.32 — (a) The null point is an unstable equilibrium (displacement of the test charge grows); (b) the midpoint of two equal like charges is an unstable equilibrium.
33. 33
    Q1.33 — y = qEL²/(2mvₓ²); identical to projectile motion under gravity.
34. 34
    Q1.34 — The electron strikes the upper plate after travelling 1.6 cm.

MCQ Questions

1 Mark Questions

1. 1
   State Coulomb's law in vector form.
2. 2
   State Gauss's law in electrostatics.
3. 3
   What is an electric dipole? Give its SI unit of dipole moment.
4. 4
   Why is the electrostatic field inside a conductor zero in equilibrium?

2 Mark Questions

1. 1
   Two charges of +3 μC and −3 μC are placed 30 cm apart in air. Find the magnitude of the force between them. (Ans: F = (1/4πε₀)q₁q₂/r² = 0.9 N)
2. 2
   An electric dipole of moment 5 × 10⁻⁹ C m is placed in a uniform field of 2 × 10⁴ N/C. Find the maximum torque on it. (Ans: τ = pE = 10⁻⁴ N m)
3. 3
   Find the electric field at a point 20 cm from a point charge of 4 μC. (Ans: E = (1/4πε₀)q/r² = 9 × 10⁵ N/C)

3 Mark Questions

1. 1
   Derive the expression for the electric field due to an infinitely long thin straight wire of uniform linear charge density using Gauss's law. (Ans: E = λ/(2πε₀r))
2. 2
   Derive the expression for the electric field due to a uniformly charged infinite plane sheet using Gauss's law. (Ans: E = σ/2ε₀)
3. 3
   Derive the electric field at a point on the axial line of an electric dipole. (Ans: E = (1/4πε₀)2p/r³)

5 Mark Questions

1. 1
   Derive the electric field of a uniformly charged spherical shell of radius R both (a) outside and (b) inside the shell using Gauss's law, and sketch the E versus r graph.
2. 2
   Describe the electric dipole — define dipole moment, derive the field on the axial and equatorial lines, and find the torque and potential energy in a uniform external field. (Ans: τ = p × E, U = −p·E)