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Getting Electric Field E from Electric Potential V (IMPORTANT) Useful because V is a scalar and superposition does not require vector sum V E ds Start by noting: B implies dV E ds A E due to a point charge (radial electric field) only nonzero term in E ds is Eradial dr dV E ds Eradial dr So: dV E Implies: radial dr ke q d ke q Check: dr r r2 (as it should be) GENERAL RELATION - ELECTRIC FIELD AND ELECTRIC POTENTIAL: E Can get components of from derivatives of V Start with: E E x iˆ E y ˆj E z kˆ and ds dxiˆ dy ˆj dz kˆ Then: dV E ds E x dx E y dy E z dz Says: dV Ex dx Ey dV dy Ez dV dz Components of E are nonzero only for directions along which V changing o If V is constant over some surface (i.e. an equipotential): E has no components along that surface Says: E is always perpendicular to equipotential surfaces E o Example: for a point charge, radial, equipotentials are spherical EQUIPOTENTIAL SURFACES: like contour maps o “valleys” centred on negative charge o “hills” centred on positive charge Example: Electric dipole (contours marked in volts) Electric field lines start on positive charge, end on negative charge Electric field lines perpendicular to equipotentials at crossing Example: Move charge q 0 from A → B 1st: What is the change in potential energy of the charge q 0 U q0 VB VA q0 24V 16V So: U q0 40 V 2nd: How much work must be done ON q 0 charge to move it from A → B? External force must do Wext q0 40 V o External force does positive work. Has to push to move positive charge from lower electric potential to higher electric potential 3rd: How much work does electric field do when q 0 is moved from A → B? Electric field does Welect U q0 40 V Along path, electric force is opposite to displacement (work negative) dV E How to understand x dx Ey E perpendicular to equipotentials everywhere. Magnitude of E is related to spacing of equipotentials. Electric field magnitude highest where equipotentials most closely spaced. Same as the relation between slope and spacing of contours on a map o Ball would roll along path perpendicular to contour o Equivalent to direction of E on equipotential map http://atlas.nrcan.gc.ca/site/english/maps/topo/map dV dy Ez dV dz Example: Calculate E by starting with electric potential V for an electric dipole Example: In some region of space, the dependence of electric potential on location is given by: V x, y, z 5 x 3 x 2 y 2 yz 2 where the coordinates are in m. (a) What is E at a point specified by coordinates (x,y,z)? E (b) What is the magnitude of at the point (1, 0, -2) ? Will now go on to talk about electric potential, electric field, and Gauss’s Law for Continuous charge distributions Strategy: Break charge distribution into small elements Sum over the effects of the carge elements Convert the sum into an integral Will then go on to electric potential and electric field near and in conductors.