ELECTRIC POTENTIAL ENERGY AND ELECTRIC POTENTIAL POTENTIAL ENERGY ELECTRIC POTENTIAL WORK-ENERGY THEOREM CAPACITANCE COMBINATIONS OF CAPACITORS STORED ENERGY Written by Dr. John K. Dayton

POTENTIAL ENERGY IN A UNIFORM ELECTRIC FIELD: In this example a charged particle is moved from point A to point B in a uniform field by the electrostatic force. The work done by the force and the change in potential energy of the particle can be calculated in the usual way. Remember, energy is a scalar quantity.

ELECTRIC POTENTIAL: Electric potential is defined as the electric potential energy per unit charge. In a uniform electric field, where  U = -qE  x, the change in electric potential will be  V = -E  x. The SI unit for electric potential is the volt, V. V=J/C. To find a particle’s change in potential energy, use: Electric potential is a scalar quantity.

EXAMPLE:What is the change in a particle’s potential energy if it moves from a position of 100 volts to a position of 150 volts? Let the particle have an electric charge of -3  C. Click For Answer

V vs r for a positive charge V r r V V vs r for a negative charge If q is a point charge (positive or negative), then the electric potential a distance r from q is given by the top equation. Note, V will be negative if q is negative. The difference between the potentials at two points is also shown. THE ELECTRIC POTENTIAL OF A POINT CHARGE:

+ - q 1 = -4  C 10cm q 2 = +3  C 15cm P r 2 =.1803m r 1 =.15m This diagram is the first step in a well planned solution. Solution continues on next slide. EXAMPLE: Calculate the electric potential on the x axis at 15.0 cm produced by two point charges; q 1 = -4.0  C on the origin and q 2 = +3.0  C at 10.0cm on the y axis. Click For Answer

Superposition of point charge potentials Electric potential is a scalar so is much easier to work with than the electric field. No directions are involved. Don’t forget to use the sign of the charge when calculating an electric potential. continued from previous slide…

A particle of charge q and mass m moves under the influence of an electric field from point A to point B. In General: THE WORK-ENERGY THEOREM FOR ELECTROSTATICS: For a single point charge: For a uniform field E: For an infinite plane of charge, : For a charged, infinite conducting sheet, :

EXAMPLE:A proton is released from rest 5.0 cm from the surface of a charged sphere of radius 10.0 cm and charge Q = 4.0  C. What will the proton’s speed be when it has moved 1 meter? Work-Energy Theorem for Electrostatics Working Equation with Potentials of a spherical charge Solved for final velocity Final Answer The proton will move between r i =.15m and r f = 1.15m. Click For Answer

CAPACITANCE AND THE CAPACITOR: A capacitor is comprised of two charged, conducting bodies maintained at a potential difference. The charge in the capacitor is actually a charge separation. Capacitance is defined as the ratio charge to potential difference within a capacitor. C = capacitance in SI units of coulombs/ volts. This combination is called the farad, F. Q = charge on capacitor. One conductor has +Q while the other has -Q. V = voltage difference between conductors. A capacitor’s capacitance depends on its size and shape, not on the charge separation and not on the voltage difference.

THE PARALLEL PLATE CAPACITOR: The parallel plate capacitor is comprised of two metal plates that face each other with each plate connected to a battery terminal. The inside area of each plate is A and they are separated by a distance d. The plate connected to the positive battery terminal will have a charge of +Q on its inside surface. The plate connected to the negative terminal will have -Q on its inside surface. The voltage difference between the plates will be the battery voltage, V. The electric field between the plate will be uniform given by pointing from the positive plate to the negative plate.

EXAMPLE:Calculate the capacitance of a parallel plate capacitor made of two circular disks of radius 6.0 cm and separated by 0.5mm. How large a radius should they have if the capacitor is to be 1.0 F? (a) (b) Click For Answer

CAPACITORS CONNECTED IN SERIES: Beginning with a group of capacitors connected in series, find the single, equivalent capacitor. In series each capacitor has the same charge on it: Q 1 = Q 2 = Q eq = Q. In series the voltages across each capacitor add to the battery voltage: V 1 + V 2 = V eq = V.

EXAMPLE:C 1 = 4.0  F and C 2 = 6.0  F are connected in series to a 24V battery. What is the stored charge in C 1 ? C 2 C V C eq + - V Use series equation Capacitors in series each have the same charge. Charge stored in C eq. Click For Answer

CAPACITORS CONNECTED IN PARALLEL: Beginning with a group of capacitors connected in parallel, find the single, equivalent capacitor. In parallel each capacitor has the same voltage across it: V 1 = V 2 = V eq = V. The individual charges add to the charge on the equivalent capacitor: Q 1 + Q 2 = Q eq.

EXAMPLE:C 1 = 4.0  F and C 2 = 6.0  F are connected in parallel to a 24.0 V battery. What is the stored charge in C 1 ? C 2 C V C eq + - V Equation for parallel Charge in Equivalent Capacitor Capacitors in parallel have the same voltage Click For Answer

CAPACITORS CONNECTED IN GENERAL: CC CeqC eqi i     11 In Parallel: In Series: C 2 C V C 3 In the diagram C 1 and C 2 are not in series; they cannot be directly combined. C 2 and C 3 are in parallel and can be combined. Once C 2,3 is known it can be combined with C 1 in series.

ENERGY STORED IN A CAPACITOR: Capacitors store energy within their electric fields. Assume the average voltage during the charging process is one-half the final voltage, V/2. Thus the charge that separates crosses this voltage and the net change in potential energy is QV/2. This is the energy stored in the capacitor: Calculate the energy density within a parallel plate capacitor: The final expression is good for any capacitor.

EXAMPLE:Calculate the equivalent capacitance of the circuit and the energy stored in each of the original capacitors. + - V C 1 = 2  F C 3 = 2  F C 2 = 3  F A series of reduced circuits. C 2 and C 3 can be combined first because they are in parallel. C 2,3 C V C eq + - V Solution Continues on Next Slide Click For Answer

continued from previous slide… Compute equivalent capacitor Distribute charge and voltage up one circuit level (Because they are in series)

continued from previous slide… (Because they are in parallel) The total of the individual stored energies must equal the energy stored in the equivalent capacitor. ANSWERSANSWERS