Line integral of Electric field: Electric Potential Electric field at a field point r, due to a point charge at origin: After the discussion of area integral of E over the surface (Flux q/ε0 ), what about Line integral of E from some point a to some other point b? And if a=∞, b=r The electric potential at a distance r from the point charge is the work done per unit charge in bringing a test charge from infinity to that point. The integral around a closed path is zero PGGC -11 DR Bhandari

Relation between E and V As fundamental theorem for gradient is Surface over which Potential is constant is called an equi-potential surface. Reference point : convention at infinity. Superposition principle: V=V1+V2+….. Unit: Nm/C or Joule/C or VOLT Curl of E ? The integral around a closed path is zero Using Stokes’ theorem => Hence curl of E In electrostatics only means no moving charge or current PGGC -11 DR Bhandari

Electric Potential due to a Point Charge Electric Field: Electric Potential: Convention: V=0 at infinite r PGGC -11 DR Bhandari

Electric Potential due to a Point Charge Electric potential at a distance r from a positive charge Q Electric potential at a distance r from a negative charge Q PGGC -11 DR Bhandari

Electric Potential due to a System of Point Charges For a system of point charges Qi at distances ri from a point P: Q1 Q4 Q3 Q2 P r1 r2 r3 r4 … an algebraic sum of scalars! PGGC -11 DR Bhandari

Electric Potential due to a Continuous Charge Distribution If the charges are not discrete but are continuously distributed over some object or region, the summation is replaced by an integral: where r is the distance from a point P where the potential is to be determined to an element of charge dq and the integral is taken over the entire distribution of charge. PGGC -11 DR Bhandari

Example of Continuous Charge Distribution: Ring of Charge Calculate the electric potential V at a distance x along the axis of a thin, uniformly charged ring of radius R carrying a total charge Q. Uniformly Charged Ring PGGC -11 DR Bhandari

Example of Continuous Charge Distribution: Uniformly Charged Disk Calculate the potential on the axis at a distance x from a uniformly charged disk of radius R. Divide the disk into thin rings of radius r and thickness dr. PGGC -11 DR Bhandari

Determination of the Electric Field from the Electric Potential If the electric potential is known in space, the electric field may be determined from it. Relationship between electric potential and electric field: The component of E in any direction is the negative of the rate of change of the electric potential with the distance in that direction. PGGC -11 DR Bhandari

Potential for a Spherical Shell of Charge Q V must be continuous, so inside the shell (r < R), On surface of sphere (r = R), PGGC -11 DR Bhandari

Potential for a Spherical Shell of Charge Outside: Inside: Inside the shell, there is no electric field, so it takes no work to move the test charge around inside the shell, therefore the electric potential is the same everywhere inside the shell. V=kQ/R PGGC -11 DR Bhandari

Electric Potential and Electric Field of a Hollow Conducting Sphere of Radius R and Net Charge Q How would the electric potential and electric field compare for a solid conducting sphere of the same radius with the same net charge? PGGC -11 DR Bhandari

Electric field at r > R: Find the potential inside and outside a uniformly charged solid sphere of radius R and total charge q. Use infinity as your reference point. Sketch V(r) . Electric field at r > R: Electric potential at r > R: R Electric field at r < R: Electric potential at r < R: PGGC -11 DR Bhandari

Equipotential Surfaces and Lines The electric potential is a scalar characteristic of the electric field (a vector quantity). Regions of space at the same electric potential are called equipotential surfaces. Point Charge Uniform Electric Field Equipotential (blue dashed) lines indicate where the equipotential surfaces intersect the page and are always perpendicular to the electric field E. PGGC -11 DR Bhandari

Equipotential surfaces are always perpendicular to electric field lines. PGGC -11 DR Bhandari

Electric Potential Energy The change ΔU in the potential energy is given by ΔU = - Wc = - ∫Fcds Relating this to a force exerted by an electric field E on a point charge qo is Fe = qo E We define the change ΔU in the electric potential energy as ΔU = - ∫ qoE ds PGGC -11 DR Bhandari

- Wc = ΔU = - ∫ qoE ds The electric potential energy depends upon the charge placed in the electric field. To quantify the potential energy in terms of only the field itself it is more useful to define it per unit charge -Wc/qo = PGGC -11 DR Bhandari

Poisson's and Laplace's Equation The electric field is related to the charge density by the divergence relationship and the electric field is related to the electric potential by a gradient relationship Therefore the potential is related to the charge density by Poisson's equation where DEL SQUARE is called Laplacian operator. In a charge-free region of space, this becomes Laplace's equation PGGC -11 DR Bhandari

Conversion from one to another Summary: Conversion from one to another ρ E V PGGC -11 DR Bhandari

Potential Difference (voltage) in terms of E: and Spherical symmetry: V = V (r) Potential energy: U = q V (joules) Equi-potential surfaces: perpendicular to field lines PGGC -11 DR Bhandari

Electric Dipole finite dipole as p = qs is the dipole moment for a finite dipole +q -q r+ r r- q s PGGC -11 DR Bhandari

Multipole expansion Legendre polynomials: coefficients of 1 expansion in powers of where and PGGC -11 DR Bhandari

Multipole expansion of V: using with PGGC -11 DR Bhandari

permittivity material Linear Dielectrics D, E, and P are proportional to each other for linear materials e 0 - permittivity space e permittivity material k e / e 0 dielectric constant ce k - 1 susceptibility PGGC -11 DR Bhandari

Capacitors (vacuum) Capacitor: charge +Q and –Q on each plate - - - - - - + + + + + + define capacitance C : Q = C DV with units 1F = 1C /1V E d PGGC -11 DR Bhandari

Energy in dielectrics vacuum energy density: dielectric: Energy stored in dielectric system: energy stored in capacitor: PGGC -11 DR Bhandari

charging a capacitor C: move dq at a given time from one plate to the other adding to the q already accumulated in terms of the field E  energy density PGGC -11 DR Bhandari

Definition of Capacitance Capacitors 1. Capacitors are devices that store electric charge 2. The capacitor is the first example of a circuit element A circuit generally consists of a number of electrical components (called circuit elements) connected together by conducting wires forming one or more closed loops Definition of Capacitance The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors C is always positive  The SI unit of capacitance is a farad (F) PGGC -11 DR Bhandari

Makeup of a Capacitor -Q +Q A capacitor is basically two parallel conducting plates with air or insulating material in between. A potential difference exists between the conductors due to the charge The capacitor stores charge E V0 d V1 PGGC -11 DR Bhandari

More About Capacitance  Capacitance will always be a positive quantity  The capacitance of a given capacitor is constant  The capacitance is a measure of the capacitor’s ability to store charge  The Farad is a large unit, typically you will see microfarads (mF) and picofarads (pF)  The capacitance of a device depends on the geometric arrangement of the conductors Metallized Polyester Film Capacitors Ceramic Capacitors Polystyrene Film Capacitors Electrolytic Capacitors Electric Double Layer Capacitors (Super Capacitors) Variable Capacitors PGGC -11 DR Bhandari

Capacitance – Isolated Sphere Capacitance – Parallel Plates Energy Stored in a Capacitor PGGC -11 DR Bhandari

Capacitors in Series C1 C2 + - C3 +Q A -Q B a b V3 V2 V1 Vab Ceq +Q -Q Substituting for V1, V2, and V3: Substituting for V: Dividing both sides by Q: Equivalent capacitance PGGC -11 DR Bhandari

Capacitors in Parallel + - V a Q3 Q2 Q1 Capacitors in Parallel Ceq + - V a Q Substituting for Q1, Q2, and Q3: Dividing both sides by V: Equivalent capacitance PGGC -11 DR Bhandari