1 Physics 2102 Gabriela González Charles-Augustin de Coulomb (1736-1806) Summary of last lecture • Electric field is a vector calculated the electric force on an imaginary 1C positive charge. • Electric field lines start or end in electric charges. • When fields are strong, electric field lines get closer. • Electric field of a single charge is |E|=kq/r2 • The “dipole moment” vector p has magnitude qa and direction from –ve to +ve charge. • Far from a dipole, |E|~kp/r3 € p http://phet.colorado.edu/en/simulation/electric-hockey2 • Approach: divide the continuous charge distribution into infinitesimally small elements • Treat each element as a POINT charge & compute its electric field • Sum (integrate) over all elements • Always look for symmetry to simplify life! Thus far, we have only dealt with discrete, point charges. In general, a charged object has a “continuous” charge distributed on a line, a surface or a volume. How do we calculate the electric field produced by such an object? ? • Useful idea: charge density • Line of charge: charge per unit length = λ"• Sheet of charge: charge per unit area = σ"• Volume of charge: charge per unit volume = ρ λ = Q/L σ = Q/A ρ = Q/V3 • Uniform line of charge +Q spread over length L • What is the direction of the electric field at a point P on the perpendicular bisector? (a) Field is 0. (b) Along +y (c) Along +x • Choose symmetrically located elements of length dx • x components of E cancel Q L a P o y x dx dx • Figure shows a uniformly charged rod of charge -Q bent into a circular arc of radius R, centered at (0,0). • What is the direction of the electric field at the origin? (a) Field is 0. (b) Along +y (c) Along -y y x • Choose symmetric elements • x-components cancel4 • Uniform line of charge, length L, total charge Q • Compute explicitly the magnitude of E at point P on perpendicular bisector • Showed earlier that the net field at P is in the y direction -- let’s now compute this! Q L a P o y x Distance Charge per unit length Q L a P o dE x dx d θ"5 What is the field E very far away from the line (L<<a)? What is field E if the line is infinitely long (L >> a)? Far away, any “localized” charge looks like a point charge Q=λL is infinite! This is not a physical situation, it is an approximation for when we are very close to the charged line (and then it “looks” infinite!). • Figure shows a uniformly charged rod of charge -Q bent into a circular arc of radius R, centered at (0,0). • Compute the direction & magnitude of E at the origin. y x 450 x y θ dQ = λRdθ dθ λ = 2Q/(πR)6 Electric field lines and forces The drawings below represent electric field lines. a) Draw vectors representing the electric force on an electron at point A, and on a proton at point B. b) If the magnitude of the force on an electron at A in (a) is 1.5µN, what is the electric field at point B in each case? We work with two different kinds of problems, easily confused: • Given certain electric charges, we calculate the electric field produced by those charges. Example: we calculated the electric field produced by the two charges in a dipole : • Given an electric field, we calculate the forces applied by this electric field on charges that come into the field. Example: forces on a single charge when immersed in the field of a dipole: (another example: force on a dipole when immersed in a uniform field)7 • Net force on dipole = 0; center of mass stays where it is. • Net TORQUE τ: INTO page. Dipole rotates to line up in direction of E. • | τ | = 2(QE)(a/2)(sin θ) "= (Qa)(E)sinθ "" = |p| E sinθ "" = |p x E| • The dipole tends to “align” itself with the field lines. • What happens if the field is NOT UNIFORM?? +Q -Q Uniform Field E Distance between charges = a θ"QE
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