35Electric Field and Potential

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ELECTRIC FIELD AND ELECTRIC FIELD INTENSITY
▢ Definition: The space around an electric charge in which its influence can be experienced is known as electric field.
▢ Electric field intensity:
❖ Definition: The electric field intensity E at any point is equal in magnitude to the force experienced per unit test position charge placed at that point and is directed along the direction of the force experienced.
❖ Formula:
◉ Basic: E = F/q or F = qE
◉ Point charge: E = (1/4πε₀) × (q/r²)
❖ Properties:
  1. Electric field intensity is a vector quantity
  2. Electric field intensity due to positive charge is always away from the charge
  3. Electric field intensity due to negative charge is always towards the charge
❖ Unit: N/C or Volt/m
❖ Resultant field: E = E₁ + E₂ + ... (vector sum)
▢ Electric Lines of Force:
❖ Definition: The line or a curve along which an isolated +Ve charge would travel if it is free to move in an electric field
❖ Properties:
  1. Start from positively charged body and end at negatively charged body
  2. No electric lines of force exist inside the charged body
  3. Tangent to the line of force at any point gives the direction of electric intensity at that point
  4. No two electric lines of force can intersect each other
  5. Always normal to the surface of a conductor, both while starting or ending
  6. Never form closed loops
  7. Always perpendicular to equipotential surface
  8. Contract longitudinally due to attraction between unlike charges
  9. Exert lateral pressure due to repulsion between like charges
  10. In uniform electric field, they are equidistant, parallel straight lines
❖ Special case: When a metallic solid sphere is placed in a uniform electric field, lines of force are normal to surface at every point but cannot pass through the conductor
▢ Electric field intensity special cases:
❖ Spheres:
◉ Spherical conductor hollow:
◈ Outside r greater than R:
■ E: E = (1/4πε) × (q/r²)
■ V: V = (1/4πε) × (q/r)
◈ Surface r equals R:
■ E: E = (1/4πε) × (q/R²)
■ V: V = (1/4πε) × (q/R)
◈ Inside r less than R:
■ E: E = 0
■ V: V = (1/4πε) × (q/R)
◈ Centre:
■ E: E = 0
■ V: V = (1/4πε) × (q/R)
◉ Non conducting sphere solid insulator:
◈ Outside r greater than R:
■ E: E = (1/4πε) × (q/r²)
■ V: V = (1/4πε) × (q/r)
◈ Surface r equals R:
■ E: E = (1/4πε) × (q/R²)
■ V: V = (1/4πε) × (q/R)
◈ Inside r less than R:
■ E: E = (1/4πε) × (qr/R³)
■ V: V = (q/4πε) × ((3R²-r²)/2R³)
◈ Centre:
■ E: E = 0
■ V: V = (q/4πε) × (3/2R)
❖ Uniformly charged ring:
◉ Axis: E = (1/4πε₀) × (qx/(R²+x²)^(3/2)) = λRx/(2ε₀(R²+x²)^(3/2))
◉ Centre: E = 0
❖ Infinite rod:
◉ Formula: E = λ/(2πrε₀)
◉ Where:
◈ λ: linear charge density
◈ r: distance from the axis of rod
❖ Non conducting infinite sheet:
◉ Formula: E = σ/(2ε₀)
◉ Where: σ = surface charge density
❖ Charged parallel plate capacitor:
◉ Formula: E = σ/ε₀
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ELECTRIC POTENTIAL AND ENERGY
▢ Electric Potential:
❖ Definition: Electric potential at a point in an electric field is defined as the work done in bringing a unit positive charge from infinity to that point
❖ Formula: V = (1/4πε₀) × (q/r)
❖ Potential difference:
◉ Definition: Work required to move a unit positive charge from point A to point B against the electric field
◉ Formula: V AB = W AB/q₀ = V B - V A
◉ Cases:
◈ W AB positive: V B > V A
◈ W AB negative: V B < V A
◈ W AB zero: V B = V A
❖ Properties:
  1. Potential due to positive charge is positive
  2. Potential due to negative charge is negative
  3. Positive charge experiences force from higher to lower potential
  4. Negative charge experiences force from lower to higher potential
  5. Work done in moving charge is independent of path (conservative field)
❖ Accelerated particle:
◉ Kinetic energy: qV = (1/2)mv²
◉ Velocity: v = √(2qV/m)
◉ KE proportionality: K.E ∝ q (at constant V)
❖ Velocity ratio: v₁/v₂ = √(q₁/q₂ × m₂/m₁)
❖ Special cases:
◉ Point charge: V = (1/4πε₀) × (q/r)
◉ Group of charges: V = V₁ + V₂ + ... + Vₙ (scalar addition)
◉ Relation with E:
◈ Formula: E = -dV/dr
◈ Note: Negative sign shows E points in direction of decreasing potential
❖ Units:
◉ E: NC⁻¹ or Vm⁻¹
◉ Note: Electric potential is scalar, but potential gradient is vector
▢ Electric Potential Energy:
❖ Definition: Total amount of work done in bringing various charges to their respective positions from infinitely large mutual separations
❖ Two charges: U = (1/4πε₀) × (q₁q₂/r)
❖ Equilateral triangle: U = (1/4πε₀) × [(q₁q₂/l) + (q₂q₃/l) + (q₁q₃/l)]
❖ Square: U = (1/4πε₀) × [(q₁q₂/l) + (q₂q₃/l) + (q₃q₄/l) + (q₄q₁/l) + (q₁q₃/√2l) + (q₂q₄/√2l)]
❖ Sign: PE may be positive or negative depending on work done against or by electric force
❖ Equilibrium: For equilibrium, PE should be zero
❖ Formula at point: P.E(U) = q × V net
❖ Where:
◉ q: charge brought from outside
◉ V net: Net potential at that point
▢ Equipotential Surface:
❖ Definition: The locus of all points which are at the same potential
❖ Properties:
  1. No work is done to move a charge from one point to another on equipotential surface
  2. Near an isolated point charge, equipotential surface is a sphere
  3. Work done to move unit positive charge around charge q along circle of radius r is zero
  4. Electric lines of force are always normal to equipotential surface
  5. Surface of charged conductor is always equipotential surface regardless of shape
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ELECTRIC DIPOLE
Image 1
Fig.Electric dipole
▢ Definition: Two equal and opposite charges separated by a small distance
▢ Dipole moment:
❖ Definition: Vector quantity whose magnitude equals product of charge and distance between charges
❖ Formula: p = q·d = q·(2l)
❖ Direction: From negative charge (-q) to positive charge (+q)
▢ Field and potential:
❖ General:
◉ E: E = (1/4πε₀) × (p√(1+3cos²θ)/r³), E ∝ 1/r³
◉ V: V = (1/4πε₀) × (pcosθ/r²), V ∝ 1/r²
◉ θ: angle made by position vector with dipole moment
❖ Axial position:
◉ E: E a = (1/4πε₀) × (2pr/(r²-l²)²)
◉ E short dipole: E a = (1/4πε₀) × (2p/r³)
◉ Angle: Angle between p and E = 0°
◉ V: V = (1/4πε₀) × (p/r²)
❖ Equatorial position:
◉ E: E b = (1/4πε₀) × (p/(r²+l²)^(3/2))
◉ E short dipole: E b = (1/4πε₀) × (p/r³)
◉ Angle: Angle between p and E = 180°
◉ V: V = 0
❖ Ratio: E a/E b = 2/1
▢ Force between dipoles:
❖ Coaxial: F = (1/4πε₀) × (6p₁p₂/r⁴)
❖ Mutually perpendicular: F = (1/4πε₀) × (3p₁p₂/r⁴)
❖ Relation: F₁ = 2F₂
▢ Dipole in uniform field:
❖ Force: F net = 0 (only rotation, no translation)
❖ Torque: τ = p × E = pE sinθ
❖ Potential energy: U = -p·E = -pE cosθ
❖ Work done: W = U₂ - U₁ = pE(cosθ₁ - cosθ₂)
❖ Special angles:
◉ θ 0:
◈ F net: 0
◈ τ: 0 (Min)
◈ U: -pE (Min)
◈ Equilibrium: Stable
◉ θ 90:
◈ F net: 0
◈ τ: pE (Max)
◈ U: 0
◉ θ 180:
◈ F net: 0
◈ τ: 0 (Min)
◈ U: pE (Max)
◈ Equilibrium: Unstable
▢ Work done revolving:
❖ Formula: W = U₂ - U₁ = pE(cosθ₁ - cosθ₂)
❖ From stable state: W ext = pE(1 - cosθ)
▢ Binding energy:
❖ Definition: Work done to separate charges of dipole to infinity
❖ Formula: U B = (1/4πε₀) × (q²/2l)
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ELECTRIC FLUX
▢ Definition: Product of electric intensity and area keeping area normal to electric intensity
▢ Formula: φ = E·A = E·A·cosθ
▢ Where: θ = angle between E and normal on area
▢ Sign:
❖ Positive: Field lines leave the area
❖ Negative: Field lines enter the area
❖ Zero: Field lines parallel to plane
▢ Type: Scalar quantity
▢ Unit: Volt-meter (Vm) or Nm²C⁻¹
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GAUSS THEOREM
▢ Statement: Total electric flux coming out of closed body equals (1/ε₀) times the net charge enclosed
▢ Formula: φ net = q net/ε₀
▢ Properties:
  1. Applicable for closed surface only
  2. Gaussian surface should always be closed
  3. Even if total flux is zero, E at Gaussian surface may be non-zero
▢ Cube with charge:
❖ Total flux: φ = q/ε₀
❖ Flux through one face: φ = q/(6ε₀)
❖ Note: φ is independent of radius of closed surface
▢ Note: Coulomb's law can be deduced from Gauss's Theorem
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FORCE PER UNIT AREA
▢ Electric pressure: dF/dA = σ²/(2ε₀) = (1/2)ε₀E²
▢ Energy density:
❖ Formula: σ²/(2ε₀) = (1/2)ε₀E²
❖ With relative permittivity: (1/2)ε₀ε rE²
▢ Direction: Always outward (±σ)² is positive
▢ Effect: Force tries to expand charged body
▢ Example: Soap bubble or rubber balloon expands on charging
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IMPORTANT FACTS
▢ Attraction of water: Thin stream of water from tap is attracted by charged rod
▢ Directions:
❖ Electric lines: From +q to -q
❖ Dipole moment: From -q to +q
▢ Work on equipotential: No work done carrying charge on equipotential surface
▢ Soap bubble: Radius increases on giving negative charge
▢ Field intensity proportional: E ∝ Number of electric lines of force
▢ Dipole in fields:
❖ Uniform: Experiences only torque, no force (torque may be zero if θ=0)
❖ Non uniform: Experiences both force and torque
▢ Point charge: No point charge produces field at its own location
▢ Charged conductor:
❖ Charge location: Resides only on outer surface
❖ Internal field: Zero at any point inside
❖ Internal potential: Constant and equal to surface potential
❖ Surface field: Proportional to surface charge density
❖ Surface potential: Independent of surface charge density
▢ Field variation:
❖ Point charge: E ∝ 1/r², V ∝ 1/r
❖ Dipole: E ∝ 1/r³, V ∝ 1/r²
❖ Quadrupole: E ∝ 1/r⁴
❖ Infinite line: E ∝ 1/r, V ∝ lnr
❖ Infinite plane: E ∝ r⁰, V ∝ r
▢ Irregular conductor: E may differ but V is same at every surface point
▢ Dipole relation: E axial = 2E broadside and V b = 0
▢ Earth potential: Zero because earth is big conductor
▢ Charged sphere: Giving positive charge decreases its mass
Q1.
Which of the following is a vector?
📅IOE 2012
Q2.
What is the potential drop across an electric hot plate which draws 5 A current when its hot resistance is 24 Ω?
📅IOM 2010
Q3.
A 2 μC charge is enclosed by a Gaussian surface of radius 0.5 m. If the radius of the Gaussian surface is doubled, the number of flux lines passing through the new surface will be
📅MOE 2068
Q4.
A sphere of radius 10 cm is charged to a potential of 300 V. The energy of the sphere is
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Q5.
The electric potential at the surface of an atomic nucleus of Z = 50 and radius 9 × 10^-15 m is
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Q6.
In a system, the electric field at the origin is along the positive x-axis. A circle centered at origin cuts axes at A(a,0), B(0,a), C(-a,0), D(0,-a). The potential is minimum at
📅KU 2012
Q7.
The energy density in the electric field created by a point charge falls off with distance r as
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Q8.
If electric field intensity is 10^6 V/m, find the distance between plates kept at a potential difference of 10^3 V.
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Q9.
NC^-1 has the same dimension as
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Q10.
An alpha particle of energy 5 MeV is scattered through 180° by a fixed uranium nucleus. The distance of closest approach is of the order of
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Q11.
Which of the following rays gets deflected by electric field?
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Q12.
When 1 C charge is moved through a potential difference of 2 V, the work done is
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Q13.
The electric field intensity between two thin metallic plates having surface charge density σ is
📅IE 2013
Q14.
Two charges -10 C and +10 C are placed 10 cm apart. The potential at the midpoint between their centers is
📅IE 2011
Q15.
Two identical charges +Q are kept at A and B. If we move from A to B, the potential
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Q16.
The magnitude of electric field E in the annular region of a charged cylindrical capacitor
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Q17.
The electric lines of force about a negative point charge are
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Q18.
A body of mass 1 kg carrying charge 1 C falls in an electric field through a potential difference of 1 V. Its velocity is
📅MOE 2009
Q19.
An electron of charge e is at rest between two plates separated by distance d and potential difference V. The force experienced by it is
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Q20.
The force between two charges in air is 10 N. On inserting dielectric, the force becomes 4 N. The dielectric constant of medium is
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Q21.
The potential difference between two charged parallel plates separated by 1 mm is 100 V. The electric field produced is
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Q22.
A strong magnetic field is applied on a stationary electron. Then the electron
Q23.
NC^-1 has the same unit as
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Q24.
Two charges +4 C and -10 C are placed at A and B 7 cm apart. The distance from A where potential is zero is
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Q25.
A charge is kept in isotropic homogeneous medium. The equipotential surface is
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Q26.
The energy per unit volume of electric field is given by
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Q27.
A charged particle of mass 1 g is placed between two charged plates of uniform field intensity 10^4 V/m. If the particle is in equilibrium, the charge on it is nearly
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Q28.
If potential of one small drop is V0 and n identical drops coalesce together, final potential of larger drop is
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Q29.
If an electron is released in electric field of strength 1 N/C, the acceleration of electron will be
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Q30.
A potential difference is applied across a wire. Then
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Q31.
If an electron is brought closer to another electron, the electric potential energy of the system
Q32.
A dielectric has strength of 10^8 V/m. The minimum voltage across a 1 mm thick specimen to puncture it is
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Q33.
A sphere is charged with charge Q. A small test charge q is placed at a distance x from the surface of the sphere. If radius is R, force on test charge is proportional to
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Q34.
Maximum value of electric intensity due to a charged sphere is at
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Q35.
An alpha particle is situated in an electric field of strength 15 × 10^4 NC^-1. The force acting on it is
Q36.
Electric field strength at distance r from point charge Q is E. If the distance of observation point is increased by 2r, electric field strength will be
Q37.
Due to the dipole shown in the figure, electric intensity will be parallel to dipole axis at point
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Q38.
In a region where electric field intensity is 5 NC^-1, 50 electric lines of force cross per square metre. Number of lines per square metre where electric field is 20 NC^-1 will be
Q39.
An electric dipole kept in a uniform electric field experiences
Q40.
Two conducting spheres of radii r1 and r2 are charged to the same surface charge density. The ratio of electric fields near their surfaces is
Q41.
Two point charges +1 μC and -1 μC are separated by 100 Å. Point P is 10 cm from midpoint on perpendicular bisector. Electric field at P is
Q42.
A solid sphere of radius R has uniform charge distribution in its volume. At distance x from centre, for x < R, electric field is directly proportional to
Q43.
The electric potential at a point on the equatorial line of an electric dipole is
Q44.
At the centre of a square ABCD, a charge is placed. The work done in moving a charge from corner A to B is
Q45.
A charge of 10 esu is placed 2 cm from a charge of 40 esu and 4 cm from another charge of 20 esu. The potential energy of 10 esu charge is
Q46.
Electric strength of air is 2 × 10^7 N/C. Maximum charge that a metallic sphere of diameter 6 cm can hold is
Q47.
A proton is about 1840 times heavier than an electron. When it is accelerated by a potential difference of 1 kV, its kinetic energy will be
Q48.
Two charges +10 μC and +50 μC are kept at a certain distance. If electric intensity at location of +10 μC is E, then electric intensity at location of +50 μC is
Q49.
Three charges +2q, -q and -q are kept at vertices of a triangle of side l. Electric intensity at centroid of triangle is
Q50.
Three charges +3q, -2q and -q are kept at vertices of an equilateral triangle of side l. The dipole moment of the system is
Q51.
A charge of 2 C is kept on axial line at some distance from centre of a dipole. If distance is doubled, the ratio of force experienced by the charge will be
Q52.
15 J work is done by an external agent in moving a charge 0.01 C from A to B. The potential difference between A and B is
Q53.
A negative charge of 1 C moves from -1000 V to +1000 V. The work done will be
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Q54.
Find the work done by external agent in moving a charge of 2 μC from infinity to a point having potential 10^5 V.
Q55.
If one penetrates a uniformly charged shell, the electric field strength
Q56.
When two electrons are brought nearer, the potential energy of the system
Q57.
Two conducting spheres of radii r1 and r2 are charged to the same surface charge density. Ratio of electric potentials on their surfaces is
Q58.
A hollow metal sphere of radius 5 cm is charged such that potential on its surface is 10 V. The potential at the centre is
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Q59.
A sphere of radius r is charged to potential V. The outward pull per unit area of its surface is
Q60.
Two insulated charged spheres of radii R1 and R2 having charges Q1 and Q2 respectively are connected to each other. There is decrease in energy of the system
Q61.
A charge +Q is kept at centre between two identical charges -q. For what value of Q/q will potential energy of the system be zero?
Q62.
A charge of 10 esu is kept 2 cm from 40 esu and 4 cm from 20 esu. Potential energy of 10 esu charge is
Q63.
A deuteron and an alpha-particle are placed in an electric field. If they are accelerated by same potential difference, velocities gained by them will be
Q64.
A particle of mass m and charge q is placed at rest in a uniform electric field E and released. The kinetic energy after moving distance y is
Q65.
Identical charges -q each are placed at 8 corners of a cube of side b. Electrostatic potential energy of a charge +q placed at centre of cube is
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Q66.
A charge q is placed at centre of a cube of side a. The electric flux through the cube is
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Q67.
Electric flux from a cube of edge l is ϕ. What is its value if edge is made 2l and charge enclosed is halved?
Q68.
A surface encloses an electric dipole. The flux through the surface is