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Equipotential surfaces and field lines Equipotential surfaces Equipotential surfaces are mathematically speaking hypersurfaces The potential in Cartesian coordinates is a function V=V(x,y,z) With V(x,y,z)=const we define a 2D surface z=z(x,y) in 3D space We can better picture the situation in 2D Example: contour lines in toporaphic maps z z( x, y ) x y 2 2 2D surface in 3D space z( x, y ) x y 2 2 2 Contour line 1D equi-value “surface” y x 2 2 Note that one could consider z ( x , y ) x 2 y 2 as equi-value surface of f ( x , y , z ) x y z 0 Some properties of equipotential surfaces -in general an equipotential surface is a hypersurface defined by V(x,y,z)=const -per definition V is the same everywhere on the surface If you move a test charge q0 on this surface the potential energy U= q0V remains constant no work done if E-field does no work along path of test charge on surface E-field normal surface E d r 0 It is a general property of the gradient of a function that f ( x , y , z ) f , f , f f ( x , y , z ) const b E dr a 2.0 x y z 1.5 f 2 x, 2 y 1.0 0.5 0.0 f ( x, y ) x y 1 Y 2 Simple 2D example for this general property f ( x, y ) x y 2 f 2 x, 2 y 2 -0.5 -1.0 -1.5 -2.0 -2.0 -1.5 -1.0 -0.5 0.0 X 0.5 1.0 1.5 2.0 2 -no point can be at different potentials equipotential surfaces never touch or intersect -field lines and equipotential surfaces are always perpendicular Examples from our textbook Young and Freedman University Physics page 799 point charge E-field lines Cross sections of equipotential surfaces How do we get the circle solution ? Remember V ( r ) Q V ( x , y , z 0) 4 0 r Q 4 0 x y 2 const 2 Q x y 4 0 const 2 2 Similar to higher field line density indicating stronger E-field Higher density of equipotential contour lines indicates a given change in the potential takes place with less distance visualization of stronger E-field because E V 2 Electric dipole Note, the electric field is in general not the same for points on an equipotential surface 2 equal positive charges Equipotentials and Conductors When all charges are at rest, the surface of a conductor is always an equipotential surface. Proof: We use the facts that i) the E-field is always perpendicular to an equipotential surface ii) E=0 inside a conductor We use ii) to show that (when all charges at rest) the E-field outside a conductor must be perpendicular to the surface at ever point With i) that implies that surface of a conductor is equipotential surface E=0 inside a conductor because otherwise charges would move E tangent to surface inside conductor zero E tangent to surface (E) outside conductor zero E=0 just outside the conductor surface to ensure Since E=0 inside conductor Remember E is conservative and work along closed path must be zero Ed r 0 in the absence of any tangential component, E, E can only be perpendicular to the conducting surface Conductor Surface of a conductor