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2/9/17 Electric Potential (2) A. B. Kaye, Ph.D. Associate Professor of Physics 9 February 2017 Update • Last class • • Finished Gauss’s Law Started Electric Potential • Today • • Continue Electric Potential Homework #2 Due • Tomorrow • • Finish Electric Potential (Ch. 25) Homework #3 Due ELECTRIC POTENTIAL Potential Difference and Potential Energy due to Point Charges 1 2/9/17 Potential and Point Charges • An isolated positive point charge produces a field directed radially outward • The potential difference between points A and B will be: Potential and Point Charges, cont. • The electric potential is independent of the path between points A and B • It is customary to choose a reference potential of V = 0 at rA = ∞ • Then the potential due to a point charge at some point r is Electric Potential of a Point Charge 2 2/9/17 Electric Potential with Multiple Charges • The electric potential due to several point charges is the sum of the potentials due to each individual charge • • This is another example of the superposition principle. The sum is the algebraic sum Electric Potential of a Dipole Potential Energy of Multiple Charges • The potential energy of the system is • If the two charges are the same sign, U is positive and work must be done to bring the charges together • If the two charges have opposite signs, U is negative and work is done to keep the charges apart 3 2/9/17 Electric Potential due to Two Point Charges, Example – –6.00 µC 3.00 m • Find the potential due to the two indicated charges at the point P + +2.00 µC P 4.00 m ELECTRIC POTENTIAL Finding the Value of the Electric Field from the Electric Potential Finding E From V • Assume, to start, that the field has only an x component • Similar statements would apply to the y– and z- components • Equipotential surfaces must always be perpendicular to the electric field lines passing through them 4 2/9/17 E and V for an Infinite Sheet of Charge • The equipotential lines are the dashed blue lines • The electric field lines are the orange lines • The equipotential lines are everywhere perpendicular to the field lines E and V for a Point Charge • The equipotential lines are the dashed blue lines • The electric field lines are the orange lines • The electric field is radial: • The equipotential lines are perpendicular to the field lines everywhere Electric Field from Potential, General • In general, the electric potential V is a function of all three dimensions • Given V (x, y, z) you can find Ex, Ey, and Ez as partial derivatives: where we have defined a new operator called the gradient • The gradient of a scalar function f is defined as the unique vector field whose dot product with any vector v at each point x is the directional derivative of f along v 5 2/9/17 The Gradient in Different Coordinate Systems • Cartesian [i.e., f(x, y, z)]: • Cylindrical [i.e., f(r, f, z)]: • Spherical [i.e., f(r, q, f)]: Electric Field from Potential Example • Given a potential that takes the form • Find the electric field ELECTRIC POTENTIAL Electric Potential due to Continuous Charge Distributions 6 2/9/17 Electric Potential for a Continuous Charge Distribution • If the charge distribution is known, we can follow this: • Consider a small charge element dq • Treat it as a point charge • The potential at some point due to this charge element is V for a Continuous Charge Distribution, cont. • To find the total potential, you need to integrate to include the contributions from all the elements • This value for V uses the reference of V = 0 when P is infinitely far away from the charge distributions V for a Continuous Charge Distribution, final • If the electric field is already known from other considerations, the potential can be calculated using the original approach: • • If the charge distribution has sufficient symmetry, first find the field from Gauss’s Law and then find the potential difference between any two points Choose V = 0 at some convenient point 7 2/9/17 V for a Uniformly Charged Ring: Example Example: • Find an expression for the electric field at a point P located on the perpendicular central axis of a uniformly charged ring of radius a and total charge Q V for a Finite Line of Charge • A rod of line ℓ has a total charge of Q and a linear charge density of l. Find the electric potential at a point P located on the y axis a distance a from the origin. 8