1 Physics 2102 Gabriela González You bring a charge of -3C from infinity to a point P on the perpendicular bisector of a dipole as shown. Is the work that you do: a) Positive? b) Negative? c) Zero? a -Q +Q -3C P2 What is the electric potential at the center of each circle? • Potential is a SCALAR • All charges are equidistant from each center, hence contribution from each charge has same magnitude: V • +Q has positive contribution • -Q has negative contribution A: -2V+3V = +V B: -5V+2V = -3V C: -2V+2V = 0 A C B -Q +Q Note that the electric field at the center is a vector, and is NOT zero for C! • Divide the charge distribution into differential elements • Write down an expression for potential from a typical element -- treat as point charge • Integrate! • Simple example: circular rod of radius R, total charge Q; find V at center. dq R3 • Uniformly charged rod • Total charge q • Length L • What is V at position P shown? P L a x dx • Electric potential: work needed to bring +1C from infinity; units = V • Work needed to bring a charge from infinity is W=qV • Electric potential is a scalar -- add contributions from individual point charges • We calculated the electric potential produced: – by a single charge: V=kq/r, – by several charges using superposition, and – by a continuous distribution using integrals.4 Notice the following: • Point charge: – E = kQ/r2 – V = kQ/r • Dipole (far away): – E ~ kp/r3 – V ~ kp/r2 • E is given by a DERIVATIVE of V! Focus only on a simple case: electric field that points along +x axis but whose magnitude varies with x. Note: • MINUS sign! • Units for E -- VOLTS/METER (V/m) • Hollow metal sphere of radius R has a charge +q • Which of the following is the electric potential V as a function of distance r from center of sphere? +q V r r=R (a) V r r=R (c) V r r=R (b)5 +q Outside the sphere: • Replace by point charge! Inside the sphere: • E =0 (Gauss’ Law) • → V=constant V E Potential inside? At r = R, V = kQ/R For r < R, V = kQ/R. • 4 point charges (each +Q) are connected by strings, forming a square of side L • If all four strings suddenly snap, what is the kinetic energy of each charge when they are very far apart? • Use conservation of energy: – Final kinetic energy of all four charges = initial potential energy stored = energy required to assemble the system of charges +Q +Q +Q +Q Do this from scratch! Understand, not memorize the formula in the book!6 • No energy needed to bring in first charge: U1=0 • Energy needed to bring in 2nd charge: • Energy needed to bring in 3rd charge = • Energy needed to bring in 4th charge = +Q +Q +Q +Q Total potential energy is sum of all the individual terms shown on left hand side = So, final kinetic energy of each charge = • Conducting surfaces are EQUIPOTENTIALs • At surface of conductor, E is normal to surface • Hence, no work needed to move a charge from one point on a conductor surface to another • Therefore, electric potential is constant on the surface of conductors. • Equipotentials are normal to E, so they follow the shape of the conductor near the surface. • Inside the conductor, E=0: therefore, potential is constant. Potential is not necessarily zero! It is equal to the potential on the surface.7 An uncharged conductor: A uniform electric field: An uncharged conductor in the initially uniform electric field: • Charge density is higher at conductor surfaces that have small radius of curvature • E = σ/ε0 for a conductor, hence STRONGER electric fields at sharply curved surfaces! • Used for attracting or getting rid of charge: – lightning rods – Van de Graaf -- metal brush transfers charge from rubber belt – Mars pathfinder mission -- tungsten points used to get rid of accumulated charge on rover (electric breakdown on Mars occurs at ~100 V/m) (NASA)8 • Electric field and electric potential: E= -dV/dx • Electric potential energy: work used to build the system, charge by charge. Use W=qV for each charge. • Conductors: the charges move to make their surface equipotentials. • Charge density and electric field are higher on sharp points of
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