advertisement

Forces and fields obey the superposition principle: Field from a group of particles is a vector sum of fields from each particle E E1 E 2 ... Ei i E x E1x E2 x ... Eix i E y E1 y E2 y ... Eiy i E z E1z E2 z ... Eiz i Electric Field Properties • A small positive test charge is used to determine the electric field at a given point • The electric field is a vector field that can be symbolized by lines in space called electric field lines • The electric field is continuous, existing at every point, it just changes in magnitude with distance from the source Electric Field Equation • Electric Field F E qo 1 qsource qsource E rˆ ke 2 rˆ 2 4 o r r • For a continuous charge distribution dq dq dE ke 2 rˆ E ke 2 rˆ r r For spatially distributed charges – we can sub-divide the object into the small, “point-like” charges and integrate (sum up) the individual fields. Often, we assign charge density for such spacious charged objects ... ... dq i dq dl for lines (1 d ) dq dA for surfaces (2 d ) dq dV for volumes (3 d ) , , correspond ing charge densities (r ) E.g. in 3 - d : E(r) ke E x ( x, y, z ) ke | r - r | 3 (r r ) dV x x (x,y,z) dx dy dz 2 2 2 3/ 2 (( x x) ( y y ) ( z z ) ) Examples of field calculations: fields of continuous charge distributions Field of a ring of charge on the symmetry axis Ex dEx E x ke dQ dQ cos k e x2 a2 r2 E Ex kQx ( x 2 a 2 )3/ 2 x x2 a2 Positive charge is uniformly distributed around a semicircle. The electric field that this charge produces at the center of curvature P is in A. the +x-direction. B. the –x-direction. C. the +y-direction. D. the –y-direction. E. none of the above Field of a disk, uniformly charged on the symmetry axis Surface charge density is , radius R Area dA of a ring of radius r dA 2 rdr; R rdr E x 2 ke x 2 2 3/ 2 (x r ) 0 E x 2 ke x x2 R2 x2 x2 r2 z dz 2rdr dz 1 x 2 R 2 2 ke x |x 2 3/ 2 z z dQ ; dQ 2 r dr dA Changing variable of integration x Ex 1 2 2 2 0 R x For the limit of x<<R, we have an electric field of the infinite plane sheet of charge and it is independent of the distance from the plane (assuming that distance x<<linear dimensions of the sheet) E plane 2 0 Electric Field Line Properties • Relation between field lines and electric field vectors: a. The direction of the tangent to a field line is the direction of the electric field E at that point b. The number of field lines per unit area is proportional to the magnitude of E: the more field lines the stronger E • Electric field lines point in direction of force on a positive test charge therefore away from a positive charge and toward a negative charge • Electric field lines begin on positive charges and end on negative charges or infinity • No two electric field lines can cross Electric field lines E is tangent to the electric field line – no 2 lines can cross (E is unique at each point) Magnitude of E is proportional to the density of the lines Remember, electric field lines are NOT trajectories! When a particle moves on a curved path, the direction of acceleration (and hence of the force) is not collinear with the tangent to the curve Electric dipole Many physical systems are described as electric dipoles – hugely important concept Water is a good solvent for ionic and polar substances specifically because of its dipole properties Torque on the electric dipole r r p qd (electric dipole moment from “-” to “+”) Electric field is uniform in space Net Force is zero Torque is not zero Net (qE )(d sin ) p E (torque is a vector) Stable and unstable equilibrium p E p E Charge #2 Three point charges lie at the vertices of an equilateral triangle as shown. Charges #2 and #3 make up an electric dipole. The net electric torque that Charge #1 exerts on the dipole is +q Charge #1 +q y –q x A. clockwise. B. counterclockwise. C. zero. D. not enough information given to decide Charge #3 Electric field of a dipole E-field on the line connecting two charges 1 1 E keq 2 2 r2 r1 - A + E r2 d 2 p ke E 3 r r1 E-field on the line perpendicular to the dipole’s axis E 2E 2 sin E2 E 2 E2 d r E 2 ke A E1 - q r2 qd E ke 3 r r r p E k e 3 r when r>>d + d General case – combination of the above two 3( p r ) p E r 3 5 r r Dipole’s Potential Energy E-field does work on the dipole – changes its potential energy Work done by the field (remember your mechanics class?) dW d pE sin d U p E Dipole aligns itself to minimize its potential energy in the external E-field. Net force is not necessarily zero in the non-uniform electric field – induced polarization and electrostatic forces on the uncharged bodies Reading Assignment : Chapter 21.5- 21.7 Next Lecture – in-class quiz about the material we covered (chapter 21 ) Homework Assignment: Check www.masteringphysics.com this afternoon Due next Tuesday by end of class