Electrostatics: Charge Density & Coulomb's Law

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Questions and Answers

Considering Coulomb's inverse square law (CISL), which statement accurately describes its applicability?

  • CISL accurately describes the force between charged plates in a capacitor.
  • CISL is applicable for point charges. (correct)
  • CISL applies universally to all charged objects, irrespective of their size or separation.
  • CISL is strictly applicable only to stationary, extended charged bodies.

Two identical conducting spheres with charges $q_1$ and $q_2$ are brought into contact and then separated. What is the charge on each sphere after separation?

  • $(q_1 + q_2)/2$ (correct)
  • $q_1$, $q_2$
  • $\sqrt{q_1 q_2}$
  • $q_1 - q_2$

A point charge q is placed at a distance r from an infinitely large grounded conducting plane. What is the magnitude of the electric field at the surface of the plane nearest to the charge?

  • $\frac{q}{16 \pi \epsilon_0 r^2}$
  • $0$
  • $\frac{q}{8 \pi \epsilon_0 r^2}$
  • $\frac{q}{4 \pi \epsilon_0 r^2}$ (correct)

A charge Q is uniformly distributed along a thin rod of length L. What is the electric potential at a point located at a distance d from one end of the rod along its axis?

<p>$V = \frac{Q}{4 \pi \epsilon_0 L} \ln(\frac{L + d}{d})$ (A)</p> Signup and view all the answers

A parallel plate capacitor is filled with two dielectrics of equal thickness but different dielectric constants $K_1$ and $K_2$ placed in series. What is the effective dielectric constant of the combination?

<p>$\frac{2 K_1 K_2}{K_1 + K_2}$ (C)</p> Signup and view all the answers

A conducting sphere of radius $R$ has a charge $Q$ uniformly distributed on its surface. What is the electric potential at a distance $r$ from the center of the sphere, where $r < R$?

<p>$\frac{Q}{4 \pi \epsilon_0 R}$ (A)</p> Signup and view all the answers

If the potential at the surface of a sphere of radius R is 100V, then what is the potential at its centre?

<p>100 V (A)</p> Signup and view all the answers

What happens to the force between two point charges if the dielectric constant of the medium between them increases?

<p>Decreases (C)</p> Signup and view all the answers

What is the electric field intensity at a point just outside a charged conductor?

<p>Perpendicular to the surface and equal to $\frac{\sigma}{\epsilon_0}$ (C)</p> Signup and view all the answers

Two capacitors, $C_1$ and $C_2$, are connected in series. The equivalent capacitance of the combination is:

<p>$\frac{C_1 C_2}{C_1 + C_2}$ (B)</p> Signup and view all the answers

The electric potential due to a small dipole at a large distance r from the dipole is proportional to:

<p>$1/r^2$ (C)</p> Signup and view all the answers

A uniform electric field is established between two parallel plates. An electron enters the field perpendicularly to the plates. What is the trajectory of the electron?

<p>Parabola (C)</p> Signup and view all the answers

What is the effect of introducing a dielectric slab between the plates of a charged parallel plate capacitor after the charging battery has been disconnected?

<p>Decreases the electric field between the plates (A)</p> Signup and view all the answers

How does the energy density in an electric field depend on the magnitude of the electric field E?

<p>Proportional to <em>E</em>^2 (A)</p> Signup and view all the answers

A capacitor is charged by a battery. The battery is then disconnected, and a dielectric slab is inserted between the plates. What quantity will decrease?

<p>Potential difference between the plates (B)</p> Signup and view all the answers

The equivalent capacitance of two capacitors in parallel is 15µF, and the capacitance of one of them is 5µF. What is the capacitance of the other capacitor?

<p>10 µF (B)</p> Signup and view all the answers

What is the amount of work done in moving a 5µC charge from point A to point B if the potential difference between these points is 20V?

<p>10^-4 J (B)</p> Signup and view all the answers

A parallel plate capacitor has a capacitance $C_0$. If the distance between the plates is doubled and the area of each plate is halved, what is the new capacitance?

<p>$C_0/4$ (D)</p> Signup and view all the answers

A dielectric material is inserted between the plates of a capacitor, which remains connected to a battery. Which of the following quantities remains unchanged?

<p>Potential difference (C)</p> Signup and view all the answers

A $2 \mu F$ capacitor is charged to 200V and then isolated. It is then connected in parallel with an uncharged $4 \mu F$ capacitor. What is the final potential difference across each capacitor?

<p>66.7 V (C)</p> Signup and view all the answers

Two point charges +q and -q are placed a distance d apart. What is the electric field at a point midway between the charges?

<p>$\frac{8kq}{d^2}$ towards -q (C)</p> Signup and view all the answers

How does the capacitance C of a spherical capacitor depend on the radius R of the sphere?

<p>C ∝ R (B)</p> Signup and view all the answers

A parallel-plate capacitor is charged and then disconnected from a battery. If the plates are then pulled apart, increasing the distance between them, what happens to the voltage across the plates?

<p>Voltage increases (B)</p> Signup and view all the answers

What is the self-energy U of a uniformly charged non-conducting solid sphere of radius R and charge q?

<p>$U = \frac{3}{8\pi\epsilon_0} \frac{q^2}{R}$ (B)</p> Signup and view all the answers

Flashcards

Charge Quantization

The quantity of charge given to or removed from a body is an integer multiple of the electron charge.

Linear Charge Density (λ)

Charge per unit length, measured along a line.

Surface Charge Density (σ)

Charge per unit area, distributed over a surface.

Volume Charge Density (ρ)

Charge per unit volume, spread throughout a 3D region.

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Coulomb's Inverse Square Law

Force between two point charges is proportional to the product of charges and inversely proportional to square of the distance.

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Electrostatic constant 'k'

The constant relating force to the product of charges divided by the square of their separation distance. k ≈ 9 x 10^9 Nm²/C²

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Neutral or Null Point

The electric field at a point where the net electric field is zero.

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Electric Potential (V)

Work done per unit charge to move a test charge from infinity to a point in an electric field.

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Capacitance

A measure of a material's ability to store electrical energy.

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Charge of a Capacitor

The charge on the positive plate of a capacitor.

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Series Combination of Capacitors

Arrangement of capacitors where the reciprocal of the equivalent capacitance is the sum of the reciprocals of individual capacitances.

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Parallel Combination of Capacitors

Arrangement of capacitors where the equivalent capacitance is the sum of the individual capacitances.

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Energy Stored in Capacitor

The electric potential energy stored in a capacitor.

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Dielectric in Capacitor

The space between capacitor plates is partially filled with a dielectric.

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Charge Redistribution

Redistribution of charge when capacitors are connected.

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Energy Loss in Capacitors

The decrease in energy when capacitors are connected.

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Capacitor-Resistor circuits

A parallel circuit of a resistor and capacitor.

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Growth of Charge

The growth of the charge within a capacitor over time.

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Decay of Charge

The decline of the charge within a capacitor over time.

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Dielectric slab force

Relationship between force and a dielectric material.

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Radius relation

Radius of a droplet is proportional to this term.

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If 'n' identical droplets each of radius 'r' are merged

The sum of identical droplets.

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Study Notes

Electrostastics

  • Charge is quantized
  • Charge given/removed is an integral multiple of an electron's charge
  • Where n = 0, 1, 2,...
  • Use formula q = ±ne
  • Where e = -1.6 x 10^-19 C

Linear Charge Density

  • This is λ
  • λ = charge/length
  • Use formula λ = q/l or λ = dq/dl
  • Therefore q = ∫ λ dl

Surface Charge Density

  • Represented by σ
  • σ = charge/area or q/A
  • Use formula σ = q/A or σ = dq/dA
  • Therefore q = ∫ σ dA

Volume Charge Density

  • use ρ
  • ρ = charge/volume
  • Use formula ρ = q/V or ρ = dq/dV
  • Therefore q = ∫ ρ dV

Coulomb's Inverse Square Law (CISL)

  • Applicable for point charges
  • Describes the force of attraction or repulsion between two point charged bodies
  • F = k(q1q2/r^2)
  • Where k = 1/(4πε0) = 9 x 10^9 Nm²/C²

Force in Free Space

  • F₀ = (1 / 4πε₀) * (q1q2 / r²)
  • ε₀ (permittivity of free space) = 8.85 x 10^-12 C²/Nm²

Force in a Medium

  • Fmed = (1 / 4πεr²) * q1q2
  • Which simplifies to: Fmed = (1/k) * (1 / 4πε₀r²) * q1q2
  • Where k is the dielectric constant
  • Also Fmed = F₀ / k

Contact Between Two Charged Bodies

  • When two charged bodies (q1, q2) touch, total charge distributes equally
  • Meaning each body has a charge of (q1 + q2) / 2
  • Fmed = F₀ / k = F/k

Dielectric Medium

  • Dielectric medium of thickness 't' taken between charges

Vector Form of CISL

  • F₂₁ = (1 / 4πε₀) * (q1q2 / r²₂₁) * r^
  • F₂₁ = -F₁₂

Charge Division for Maximum Force

  • If a charge Q divides into two equal parts separated by a distance, the force between them is maximized

Intensity of Electric Field

  • E = F/q₀
  • Where F is force on test charge q₀

Electric Field Due to Point Charge

  • E = (1 / 4πε₀) * (q / r²)

Neutral or Null Point

  • Point where electric field is zero
  • For like charges, it lies between them

Neutral Point Location

  • x = r / (√(q₂/q₁) + 1)

Unlike Charges

  • For unlike charges, the neutral point is outside them

Acceleration in Uniform Electric Field

  • a = qE / m

Simple Pendulum in Electric Field

  • Upward field: T = 2π√(l / (g - Eq/m))
  • Downward field: T = 2π√(l / (g + Eq/m))
  • Horizontal field: T = 2π√(l / √(g² + (Eq/m)²))

Electric Potential

  • Work to bring test charge q₀ from infinity to a point
  • V = W / q₀
  • AV = -∫E · dr
  • Scalar quantity

Potential Due to Point Charge

  • V = (1 / 4πε₀) * (q / r)

Potential Due to System of Charges

  • V = (1 / 4πε₀) * (q₁/r₁ + q₂/r₂ + q₃/r₃ + ...)

Potential Energy of Charges

  • For two charges: U = (1 / 4πε₀) * (q1q2 / r)
  • For three charges: U = (1 / 4πε₀) * (q1q2/r12 + q2q3/r23 + q3q1/r31)
  • System at equilibrium has zero net electrostatic potential energy

Uniformly Charged Thin Rod/Wire

Total Electric Field at Point P

  • Along x-axis: Ex = λ / (4πε₀r) * (sin α + sin β)
  • Along y-axis: Ey = λ / (4πε₀r) * (cos β - cos α)

Resultant Field

  • Enet = √(Ex² + Ey²)
  • tan φ = Ey / Ex

Infinite Charged Wire

  • α = β = π/2
  • Ex = λ / (2πε₀r)
  • Ey = 0

Infinite Length Wire (One End)

  • α = π/2, β = 0
  • Ex = λ / (4πε₀r)
  • Ey = λ / (4πε₀r)
  • E = Ex = Ey, and φ = 45°

Uniformly Charged Circular Arc

  • E = (1 / 4πε₀) * (q / R²) * sin(Φ/2) / (Φ/2)
  • For semicircle: Φ = π

Uniformly Charged Ring

  • E = qx / (4πε₀(R² + x²)^(3/2))
  • Field at the center: Ec = 0
  • V = q / (4πε₀√(R² + x²))

Uniformly Charged Disc

  • E = σ / (2ε₀) * (1 - x / √(R² + x²))
  • For infinite disc: E = σ / (2ε₀)
  • V = σ / (2ε₀) * (√(R² + x²) - x)

Electric Flux

  • φ = E · A = EA cos θ
  • Scalar
  • Inward flux is negative, outward is positive

Gauss Law

  • Net flux through closed surface is Φnet = Qen / ε₀

Uniformly Charged Cylinder

  • Conducting: E = σR / (ε₀r)
  • Non-conducting: E = ρr / (2ε₀)

Uniformly Charged Sheet

  • Conducting: E = σ / ε₀
  • Non-conducting: E = σ / (2ε₀)

Self-Energy

  • U = (1 / 8πε₀) * (q² / R)

Electric Dipole

  • p = q(2l)
  • Direction: -q to +q

Electric Potential Due to Dipole

  • V = (1 / 4πε₀) * (p cos θ / r²)

Electric Field Due to Dipole

  • Er = (1 / 4πε₀) * (2p cos θ / r³)
  • Eθ = (1 / 4πε₀) * (p sin θ / r³)

Torque

  • τ = p × E

Potential Energy

  • U = -pE cos θ

Time Period

  • T = 2π√(I / pE)

Force Between Dipoles

  • F = (1 / 4πε₀) * (3p₁p₂ / r⁴)

Electrostatic Pressure

  • P = σ² / (2ε₀)

Capacitance

  • C = q / V

Spherical Capacitor

  • C = 4πKε₀ * (ab / (b - a))

Cylindrical Capacitor

  • C = 2π εL / ln(b/a)

Capacitor Resistance Circuit

  • Growth of charge: q = q₀(1 - e^(-t/CR))
  • Decay of charge: q = q₀(e^(-t/CR))

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