Potential Due To Different Charge Distributions

An Introduction

• Electric potential: measure of electrical potential energy per unit charge
• Scalar quantity: defined as the amount of work done per unit charge to move it from a reference point to a specific location
• Common reference point is infinity, where potential is zero
• Electric potential difference: change in potential energy experienced by a charge when it moves between two points

Electric Potential Difference (V)

• Symbol: V
• Unit: Volt (V)
• Equation: V = $ \dfrac{W}{q} $ , where W is work done on moving the charge and q is the charge
• Denoted by VAB, representing potential at point A subtracted by potential at point B

Relation between Electric Potential and Electric Field

• Electric field (E) at a point is the force experienced per unit positive charge at that point
• Electric field and electric potential are closely related:
  • ΔV = - $ \int $ $ \overrightarrow{E} $ $ \bullet $ $ \overrightarrow{dr} $
  • Electric field is the negative gradient of electric potential: $ \overrightarrow{E} $ = - $ \nabla V $

Potential Due to a Point Charge

• Potential due to a point charge (q) at a distance (r) from it:
  • V = $ \dfrac{kq}{r} $
  • k is the Coulomb’s constant ( $ 9 \times 10^9 , \text{Nm}^2/\text{C}^2 $ )
• As distance increases, potential decreases and approaches zero at infinity

Potential Due to a System of Point Charges

• For a system of point charges, the total potential is the algebraic sum of potentials due to individual charges
• $ \sum V = \sum \dfrac{kq_i}{r_i} $ , where q_i is the charge and r_i is the distance of the i-th charge from the point

Potential Due to Uniformly Charged Sphere

• For a uniformly charged sphere, potential at a point inside or outside the sphere:
  • V = $ \dfrac{3kQ}{2R} $ , where Q is the total charge and R is the radius of the sphere
• Inside the sphere, the potential is directly proportional to the distance from its center
• Outside the sphere, the potential follows the inverse square law

Potential Due to an Electric Dipole

• Electric dipole: a pair of equal and opposite charges seperated by a small distance
• Potential at a point due to an electric dipole:
  • V = $ \dfrac{kqd}{r^2}\cos(\theta) $ , where q is the charge, d is the distance between the charges, r is the distance from the dipole, and θ is the angle between the dipole axis and the line joining the charges to the point

Equipotential Surfaces

• Equipotential surfaces: surfaces on which the potential is constant and do not intersect
• Electric field and equipotential surfaces are perpendicular to each other
• Electric field lines cross equipotential surfaces at right angles
• Electric potential difference between two equipotential surfaces is zero

Potential Due to Continuous Charge Distribution

• For continuous charge distributions, potential at a point:
  • V = $ \int \dfrac{kdQ}{r} $ , where dQ is the charge element and r is the distance from the element to the point
• Examples of continuous charge distributions:
  • Line of charge
  • Charged disk or ring
  • Charged conducting shell

Potential Difference in a Capacitor

• Capacitor: device used to store electric charge and energy
• It consists of two conductive plates separated by a dielectric material
• Equation for potential difference in a capacitor:
  • V = $ \dfrac{Q}{C} $ , where Q is the charge stored and C is the capacitance
• Capacitance (C) depends on the size and shape of the capacitor, as well as the dielectric constant of the material used

Electric Potential Due to a Line of Charge

• Line of charge: a long, thin charged object
• Electric potential due to a line of charge:
  • V = $ \dfrac{2k\lambda}{r} $ , where λ is the charge density (charge per unit length) and r is the distance from the line of charge
• The potential decreases as the distance from the line of charge increases

Electric Potential Due to a Charged Disk or Ring

• Electric potential due to a charged disk at a point on its axial line:
  • V = $ \dfrac{\sigma}{2\epsilon_0}(1 - \dfrac{1}{\sqrt{1 + (z/R)^2}}) $ , where σ is the surface charge density and z is the distance of the point from the center of the disk
• Electric potential due to a charged ring at a point on its axial line:
  • V = $ \dfrac{kQ}{\sqrt{R^2 + z^2}} $ , where Q is the total charge of the ring and R is the radius of the ring

Electric Potential Due to a Charged Conducting Shell

• For a charged conducting shell, the potential is the same at all points inside the shell
• Potential inside the conducting shell:
  • V = $ \dfrac{kQ}{R} $ , where Q is the charge of the shell and R is the radius of the shell
• Potential outside the conducting shell follows the inverse square law

Electric Potential Energy

• Electric potential energy: the energy possessed by a charge due to its position in an electric field
• Electric potential energy of a charge (q) at a point with potential (V):
  • U = qV
• Positive work is done on a charge when it moves from a lower potential to a higher potential
• Negative work is done when the charge moves from a higher potential to a lower potential

Relationship Between Electric Potential and Electric Field

• Electric potential and electric field are related through the equation:
  • E = - $ \dfrac{dV}{dr} $ , where E is the electric field, V is the electric potential, and r is the distance from the point
• Electric field lines point in the direction of decreasing potential
• Equipotential surfaces are perpendicular to electric field lines

Potential Difference in an Electric Circuit

• Electric circuits: closed paths through which electric current flows
• Potential difference (voltage) across a component in an electric circuit:
  • V = IR, where I is the current flowing through the component and R is its resistance
• Ohm’s Law: V = IR, which states that the current through a conductor is directly proportional to the potential difference across it, provided its temperature remains constant

Potential Difference in Series and Parallel Circuits

• In a series circuit, the potential difference across components adds up:
  • V_total = V_1 + V_2 + V_3 + …
• In a parallel circuit, the potential difference across components is the same:
  • V_total = V_1 = V_2 = V_3 = …

Capacitors and Capacitance

• Capacitor: a device used to store electric charge and energy
• Capacitance (C): a measure of a capacitor’s ability to store charge
  • C = $ \dfrac{Q}{V} $ , where Q is the charge stored and V is the potential difference across the capacitor
• Unit of capacitance: Farad (F), where 1F = 1C/V
• Capacitance depends on the size, shape, and dielectric constant of the capacitor

Capacitors in Series and Parallel

• Capacitors in series:
  • $ \dfrac{1}{C_{\text{total}}} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \dfrac{1}{C_3} + … $
• Capacitors in parallel:
  • $ C_{\text{total}} = C_1 + C_2 + C_3 + … $
• Capacitors in series share the same charge, while capacitors in parallel share the same potential difference

Energy Stored in a Capacitor

• Energy stored in a capacitor (U):
  • U = $ \dfrac{1}{2}CV^2 $ , where C is the capacitance and V is the potential difference across the capacitor
• Energy is stored in the electric field between the capacitor plates
• Energy density (u) is the energy stored per unit volume
  • $ u = \dfrac{1}{2}\epsilon_0 E^2 $ , where ε₀ is the permittivity of free space and E is the electric field between the plates

Electric Potential and Work Done

• Electric potential is related to the work done in moving a charge:
  • Work done (W) = qΔV, where q is the charge and ΔV is the change in electric potential
• Positive work is done when a charge is moved in the direction of increasing potential
• Negative work is done when a charge is moved in the direction of decreasing potential

Equipotential Lines

• Equipotential lines: lines on which the electric potential is constant
• Equipotential lines are always perpendicular to electric field lines
• Equipotential lines are closer together where the electric field is stronger
• Equipotential lines never intersect

Electric Potential Inside a Conductor

• Inside a conductor in electrostatic equilibrium, the electric field is zero
• Therefore, the electric potential is constant throughout the conductor
• Charges on a conductor reside on its surface, redistributing until the potential is constant
• Charges move until the electric field inside the conductor is zero

Capacitance of a Parallel Plate Capacitor

• A parallel plate capacitor consists of two parallel plates with equal and opposite charges
• Capacitance (C) is a measure of the capacitor’s ability to store charge
• Capacitance of a parallel plate capacitor:
  • $ C = \dfrac{\epsilon_0 A}{d} $ , where ε₀ is the permittivity of free space, A is the area of each plate, and d is the distance between the plates
• The greater the area and smaller the distance, the higher the capacitance

Energy Stored in a Parallel Plate Capacitor

• Energy stored in a parallel plate capacitor (U):
  • $ U = \dfrac{1}{2} CV^2 $ , where C is the capacitance and V is the potential difference across the plates
• The energy is stored in the electric field between the plates
• The energy density ( $ u $ ) is the energy stored per unit volume:
  • $ u = \dfrac{1}{2}\epsilon_0 E^2 $ , where E is the electric field strength between the plates

Dielectrics in Capacitors

• Dielectric: insulating material placed between the plates of a capacitor
• Dielectrics increase the capacitance of a capacitor:
  • $ C_{\text{with dielectric}} = kC_{\text{without dielectric}} $ , where k is the dielectric constant
• Dielectrics reduce the electric field strength between the plates
• Energy stored in a capacitor with a dielectric:
  • $ U_{\text{with dielectric}} = \dfrac{1}{2} kCV^2 $

Capacitors in an Electric Field

• Capacitors can also be used to create electric fields
• A charged capacitor can exert a force on other charges
• The electric field created by a charged capacitor:
  • $ E = \dfrac{\sigma}{\epsilon_0} $ , where σ is the surface charge density of the plates and ε₀ is the permittivity of free space

Applications of Capacitors

• Capacitors have various applications in electronic circuits:
  1. Energy storage in power supplies
  2. Filtering out noise and stabilizing voltage levels
  3. Timing and oscillation circuits in electronic devices
  4. Energy storage in camera flashes
  5. Motor start capacitors in appliances

Electric Potential and Potential Difference

• Electric potential: the amount of electric potential energy per unit charge at a point in an electric field
• Potential difference: the change in electric potential between two points in an electric field
• The potential difference between two points is equal to the work done per unit charge in moving a charge between those points
• Electric potential difference is measured in volts (V)

Summary

• Electric potential is the measure of electrical potential energy per unit charge
• Potential due to point charges, continuous charge distributions, and capacitors can be calculated using appropriate formulas
• Equipotential surfaces are surfaces on which the potential is constant and do not intersect
• Capacitance is a measure of a capacitor’s ability to store charge and energy
• Capacitors in series and parallel have different total capacitances
• Energy is stored in capacitors and is related to the capacitance and potential difference across the capacitor