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The Electric Potential 28.1 Electric Potential Energy A) Items to recall A) Conservative forces (1) Allow us to figure changes in energy levels B) Any change in position is final minus initial C) Work can be calculated by Force times distance, but what if force is non-constant B) Uniform Fields A) Our first analogy to gravitation (1) The gravitational field near the surface of the earth is uniform π = 9.8 m/s2 (2) An object moving in the direction of the gravitational field looses energy based on its mass, the field strength, and the distance moved (3) πππππ£ = π0 + πππ¦ B) Charges moving in an electric field will have similar losses and gains in electric potential energy (1) βπππππ = ππ β ππ = βπππππ (π β π) = ππΈπ π β ππΈπ π (2) πππππ = π0 + ππΈπ C) Decreases and Increases in Potential Energy (1) Negative and positive charges influence gain or loss (2) Direction of motion with or against field influence gain or loss 28.2 The Potential Energy of a Point Charge A) Establishing the equation π₯ π₯ πΎπ1 π2 ππ₯ π₯2 π A) πππππ = β«π₯ π πΉ1on2 ππ₯ = β«π₯ π π B) βπππππ = ππ β ππ = βπππππ (π β π) = C) πππππ = πΎπ1 π2 π₯ = β1 π₯π | = π₯ π₯π πΎπ1 π2 πΎπ π β π₯1 2 π₯π π = πΎπ1 π2 β πΎπ1 π2 π₯π + πΎπ1 π2 π₯π 1 π1 π2 4ππ0 π The choice π0 = 0 is equivalent to saying that the potential energy of two charged particles is zero when they are infinitely far apart - Potential energy for two like charges is positive and for two opposite charges is negative, the positive and negative charges want to be next to one another. - The equation also works for spheres of charge since all the charge can be considered at the center of the sphere - βFar apartβ or βfar awayβ can mean πππππ β 0 B) Electric Force Is a Conservative Force A) Work done is independent of the path taken to get from initial to final position C) Multiple Point Charges - A) πππππ = βπ<π 28.3 πΎππ ππ πππ , stipulation π < π, describes that each charge is counted only once The Potential Energy of a Dipole A) An electric dipole placed in a field has the opportunity to begin moving, rotationally, it has potential energy A) this was discussed previously when the torque on the dipole was found. π = πβ × πΈββ B) ππππππ = βππΈ sin πππ π C) πππππ = βππΈ β«π π sin π ππ = ππΈ cos ππ β ππΈ cos ππ π D) Ξπdipole = ππ β ππ = βπππππ (π β π) = βππΈ cos ππ + ππΈ cos ππ E) πdipole = βππΈ cos π = βπβ β πΈββ F) Torques, we want to find the force perpendicular to lever arm; Work, we want to find force parallel to displacement 28.4 The Electric Potential A) Property of a source charge A) A source charge will create an electric field B) Placing another charge in that field will allow (1) There to be a force on both the charges (2) There to be a potential energy established C) The amount of electric potential energy per charge in the field π πβ‘ π Joule D) Units are Coulomb = Volt B) Using the Electric Potential A) Often the difference in two electric potentials is more important than the value of a single electric potential (1) βπ = ππ β ππ (2) This is called electric potential difference or voltage B) Conservative forces are at work, so conservation of energy can help to determine many factors about the motion of a charge through a potential difference (1) πΎπ + πππ = πΎπ + πππ (2) Table 28.2 C) Example 28.6 28.5 The Electric Potential Inside a Parallel-Plate Capacitor - Normally begin with values relative to a point charge, electric force, electric field, electric flux, however, Electric Potential studies begin with a capacitor because the electric field between the plates is uniform I) Electric Potential Energy of the Capacitor A) πππππ = π0 + ππΈπ , where ππ = 0 1) πππππ = ππΈπ , β΄ 2) π = πΈπ a) The electric potential exists at all points between the plates b) Electric potential increases linearly from negative plate c) π is the distance from the negative electrode B) Potential difference of a capacitor 1) βππΆ = π+ β πβ = πΈπ 2) Rearranging the equation allows us to get πΈ = βππ , π allowing us to discover new units for electric field a) 1 N/C = 1 V/m C) The electric field within a capacitor is constant, but not the potential π 1) π = πΈπ = π βππΆ II) Equipotential Surfaces A) Lines that detail points of like potential B) Analogous to lines on topographical map C) Lines of e.p. will be perpendicular to electric field lines D) Usually established in natural segments of voltage, i.e. 5V, 10V, 15V or 2V,4V,6V III) Battery establishes the Charge on the Plates 28.6 I) The Electric Potential of a point Charge Establishing the equation 1 πβ²π 4ππ0 π ππβ² & π 1 π = πβ² 4ππ0 π A) ππβ² & π = B) π = II) Visualizing the Potential of a Point Charge III) **Note both similarities and differences of this figure and the drawings done in the special section βGraphical representation of the electric potential inside a capacitorβ** The Electric Potential of a Charged Sphere π A) π = πΎ π , (sphere of charge, π β₯ π ) B) π = 28.7 I) π π π 0 (sphere charged to potential π0 ) The Electric Potential of Many Charges Superposition 1 A) π = βπ 4ππ II) 0 ππ ππ B) Principle of Superposition says the total is the sum of the individuals, benefit that potential is a scalar quantity Continuous Distribution of Charge A) PSS 28.2