Electric Dipoles: Potential Energy

• Electric field intensity between two parallel metallic plates with electric potentials ( V_1 ) and ( V_2 ) separated by distance ( d ) is ( E = \frac{V_2 - V_1}{d} ) in V/m.
• Electric potential energy of two electric dipoles: ( U = -\frac{1}{4\pi\epsilon_0} \frac{2p_1p_2}{r^3} )

Potential Energy of a Dipole

• Potential energy of a dipole in an external field is equal to the work done in rotating the dipole from angle ( \theta_0 ) to ( \theta_1 ) with respect to the electric field ( E ).
• The work done by the external torque is given by ( W = \int_{\theta_0}^{\theta_1} \tau d\theta = \int_{\theta_0}^{\theta_1} pE \sin\theta d\theta ).
• ( W = pE [-\cos\theta]_{\theta_0}^{\theta_1} = pE (\cos\theta_0 - \cos\theta_1) ).
• The work done is stored as the potential energy of the system.
• Potential energy of the dipole placed in an external field ( E ) is ( U(\theta) = pE (\cos\theta_0 - \cos\theta_1) )

Particular Cases

• When the dipole is initially aligned along the electric field, i.e., ( \theta_0 = 0^\circ ), and is set at angle ( \theta ) with ( E ):
  • ( W = -pE (\cos\theta - \cos 0^\circ) = -pE (\cos\theta - 1) )
  • The work is stored in the dipole as potential energy.
• When the dipole is initially perpendicular to ( E ), i.e., ( \theta_0 = 90^\circ ), and set at angle ( \theta ) with ( E ):
  • ( W = -pE (\cos\theta - \cos 90^\circ) = -pE \cos\theta )
  • Potential energy of dipole is ( U = W = -pE \cos\theta ) or ( U = -p \cdot E ).
• Potential energy of an electric dipole is a scalar quantity, measured in joules.

Conductors and Insulators

• Conductors allow electric charge to flow easily (e.g., metals, silver).
• Insulators inhibit the flow of electric charge (e.g., glass, rubber, wood); also called dielectrics.
• Applying an electric field to dielectrics induces charges on the surface.
• Dielectrics transmit electric effects without conducting.

Polarization

• In a dielectric, an electric field produces a net dipole moment, known as dielectric polarization.

Capacitor

• A capacitor stores electric charge without changing its shape
• A capacitor consists of two conductors separated by an insulating medium.

Capacity

• Capacity is the measure of a capacitor's ability to store charge and electrical energy.
• Capacitance is the ratio of charge given to the capacitor to the increase in its potential: ( q = CV ), where ( C ) is capacitance.
• Capacitance depends on shape, size, separation between conductors, surrounding medium, and presence of other conductors.
• SI unit of capacitance is the farad (F); its dimensional formula is ( [M^{-1}L^{-2}T^4A^2] ).
• 1 farad = 1 coulomb / 1 volt
• Farad is a large unit; microfarads (( \mu )F) and picofarads (pF) are common, with ( 1\mu F = 10^{-6} F ) and ( 1pF = 10^{-12} F ).
• Capacitance of an isolated spherical capacitor: ( C = 4\pi\epsilon_0 a ), where ( a ) is the radius of the sphere.

Common Potential

• When two capacitors with different potentials are connected by a wire, charge flows from higher to lower potential until they reach a common potential.
• Common potential is the total charge divided by the total capacitance: ( V = \frac{C_1V_1 + C_2V_2}{C_1 + C_2} ).
• Loss of energy during charge redistribution: ( \Delta U = \frac{C_1C_2(V_1 - V_2)^2}{2(C_1 + C_2)} )

Dielectric Strength and Breaking Potential

• Dielectric strength is the maximum electric field a dielectric can withstand without breakdown; for air, it is about ( 3 \times 10^6 ) V/m.
• Breaking potential is the maximum potential applied on a dielectric without it breaking down.

Parallel Plate Capacitor

• It comprises two metal plates of area A separated by distance d, filled with air or a dielectric medium.
• Capacitance of an air-filled parallel plate capacitor: ( C_0 = \frac{\epsilon_0 A}{d} ).
• With a dielectric of constant ( K ) fully filling the space: ( C = \frac{K\epsilon_0 A}{d} = KC_0 ).
• With a dielectric slab of thickness ( t ) inserted: ( C = \frac{\epsilon_0 A}{d - t + \frac{t}{K}} )