Potential of a Uniformly Charged Disk

• The potential at a point P with continuous charge distribution is the sum of individual differential charge elements dq contributions
• With infinity as a zero-potential reference the electric potential at P due to dq is dV = (1 / 4πε₀) * (dq / r)
• The total potential over all differential elements is then V = ∫dV = ∫(1 / 4πε₀) * (dq / r)

Electric Potential on the Axis of a Charged Ring

• For a uniformly charged ring of radius a and charge Q in the z = 0 plane, centered at the origin, the distance from a charge element dq to field point P on the ring's axis is r = √(z² + a²)
• The potential at a point due to the ring is obtained by integrating: V = ∫(k * dq / r) = (k / r) ∫dq = kQ / r = kQ / √(z² + a²)
• When |z| is much greater than a, the potential approaches kQ / |z|, similar to a point charge Q at the origin

Electric potential V for a Charged Disk

• Potential on the axis of a disk with radius R and total charge Q distributed uniformly is found by treating the disk as rings
• A ring with radius a and thickness da in Fig. (8) has an area of dA = 2πa da
• The charge of the ring is dq = σdA = σ2πa da, where σ = Q / πR² (surface charge density)
• Potential at point P due to ring charge is dV = kdq / √(z² + a²)
• Integrate from a = 0 to a = R to find the total potential V = ∫₀ᑅ (kσ2πa da) / √(z² + a²) = kσπ[√(z² + R²) - |z|]
• Rearranged, V(z) = 2kσπ|z| * [√(1 + R² / z²) - 1]
• The potential on the axis of a uniformly charged disk is V(z) = 2kσπ|z| * [√(1 + R² / z²) - 1]

Electric field from Electric Potential

• The relation between E and V for two points with small distance ds apart is dV = -E · ds
• In Cartesian coordinates: E = Exî + Eyĵ + Ezk and ds = dx î + dy ĵ + dz k
• Therefore: dV = -(Exî + Eyĵ + Ezk) · (dx î + dy ĵ + dz k) which simplifies to dV = Exdx + Eydy + Ezdz
• Also: dV = (∂V/∂x)dx + (∂V/∂y)dy + (∂V/∂z)dz
• Which means: Ex = -∂V/∂x, Ey = -∂V/∂y, Ez = -∂V/∂z
• The electric field can be expressed: E = Exî + Eyĵ + Ezk = -(∂V/∂x)î - (∂V/∂y)ĵ - (∂V/∂z)k
• Mathematically think of Electric Field as the negative gradient of the electric potential V
• Physically, when V increases as a positive charge moves along x, there is a non-vanishing component of E in the x opposite direction (-Ex ≠ 0)
• For charge distribution with spherical symmetry, E is a function of radial distance r: E = Eᵣr̂
• In this case, dV = -Eᵣdr, and if V(r) known, E = Eᵣr̂ = -(dV/dr)r̂
• Example: electric potential due to point charge q is V(r) = 1/(4πε₀) * (q/r), and E = 1/(4πε₀) * (q/r²)r̂

Examples

• For a potential function V(x) = 100 V - (25 V/m)x, because V only depends on x, Ex = -dV(x)/dx = 25 V/m, and Ey = Ez = 0
• For a ring with radius a and charge Q, potential at point P on the ring axis a distance x is V(x) = 1/(4πε₀) * (Q/√(x² + a²))
  • The x-component of the electric field is Ex = Qx / [4πε₀(x² + a²)³/²]
• Electric potential due to a charge distribution can be written as V(x, y, z) = Ax²y² + Bxyz where A and B are constants
  • Electric field components are found by taking partial derivatives: Ex = -(2Axy² + Byz), Ey = -(2Ax²y + Bxz) and Ez = -Bxy
  • The electric field: E(x, y, z) = -(2Axy² - Byz)î - (2Ax²y + Bxz)ĵ - (Bxy)k

Equipotential Surfaces and Field Lines

• A system in 2D with electric potential V(x, y), equipotential curves have constant V(x, y)
• In 3D, equipotential surfaces are described by V(x, y, z) = constant
• The potential V is constant on an equipotential surface, and for a small displacement dl parallel: dV = -E · dl = 0
• E must be zero or perpendicular to every dl parallel to the surface, meaning E is normal to the surface
• Electric field lines are normal to equipotential ones they intersect
• The field lines in the plane of the charges are represented by red lines, intersections by blue lines
• Equipotential surfaces are 3D, at each crossing of equipotential/field line, the two are perpendicular

Equipotential properties

• Electric field lines are perpendicular pointing from high to low potentials
• Equipotential surfaces from point charge form concentric spheres; constant electric field forms planes perpendicular
• The tangential component of Electric field along the equipotential surface is zero, no work is done while moving charge
• No work needed for a particle along equipotential surface