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PHY481: Electrostatics Introductory E&M review (2) Course web site: www.pa.msu.edu/courses/phy481 All homework is due AT 12:30 pm (no later) HW1 due one week from today Lecture 2 Carl Bromberg - Prof. of Physics Field of a charged spherical shell - Gauss’s Law Consider a radius R spherical shell with surface charge density σ. – A spherical Gaussian surface with radius r < R, is inside the charge surface, and encloses no charge -> Einside = 0. – A spherical Gaussian surface with radius r > R, has the electric field normal to its surface. $ 2$ 2 E ! dA = E r "S " sin# d# " d% 0 0 q 2 E4$ r = &0 q = ! 4" R 2 q q E= ; E= r̂ 2 2 4$& 0 r 4$& 0 r Same electric field as charge q at the origin Lecture 2 Carl Bromberg - Prof. of Physics 1 Uniform charge density sphere - Gauss’s Law Find the electric field inside of a sphere, radius R, with a uniform charge density ρ throughout the volume. – Pick a spherical Gaussian surface, radius r, inside the charged sphere $ 2$ 2 E ! dA = Er "S " sin# d# " d% 0 0 3 q Qr encl E4$ r 2 = = &0 & 0 R3 !=Q/ Qr Qr E= ; E= r̂ 3 3 4$& 0 R 4$& 0 R qencl = ! ( ( 3 4 " R 3 4 "r3 3 ) ) r3 =Q 3 R – Electric field outside of a sphere E= Lecture 2 Q r̂ 2 4!" 0 r Same electric field as charge Q at the origin Carl Bromberg - Prof. of Physics 2 Infinite sheet (again) - Gauss’s Law Infinite sheet of charge with surface density σ. – Pick a cylindrical Gaussian surface, radius r, passing through the sheet. – The dot product E•dA is non zero only on TWO the ends. R 2$ "S E ! dA = 2E " rdr " 0 qencl 2(E$ R ) = %0 2 qencl = ! (" R 2 ) d# 0 & ($ R 2 ) = %0 & & E= ; E=± k̂ 2% 0 2% 0 Lecture 2 + above – below Carl Bromberg - Prof. of Physics 3 Electric Potential Energy U(x) Potential energy change ΔU of charge q’ in known E-field. – Calculate the (negative) of the work done by the field along a path r2 r2 U is a scalar ! !U = " $ Felec # ds = " q% $ E # ds r1 r1 Example: parallel plate capacitor x !U = q"E # dx " = q ' Ex 0 E = ! E î ds = dx " î Move q’ across entire capacitor !U = U (d) " U (0) = q ' Ed Lecture 2 U = 0 at negative plate. Carl Bromberg - Prof. of Physics 4 Electric potential V(x) Potential V(x) is potential energy/unit test charge q’ – The potential V is defined without the test charge q’ PE of test charge U&V !U =# are scalars ! !V = q" r2 % E $ ds r1 U (x) + arbitary constant V (x) = q! Example: parallel plate capacitor Set V(0) = 0 constant = 0 U (x) V (x) = = Ex V (d) = Ed q! Equipotential surfaces – Electric field lines always cross equipotential surfaces at 90°. – In parallel plate capacitors equipotential surfaces are planes parallel to plates Lecture 2 Carl Bromberg - Prof. of Physics 5 Batteries, capacitors, and energy storage A battery moves charge Q between plates of area A – Battery moves electrons to create charge densities σ. – We have two expressions for electric field E ! E= ! E= "0 VB d – Find expression relating Q and V !0 A Q= VB = CVB d != Q A !0 A C= d – Find energy stored while charging plates to Q U = ! dU = Lecture 2 Q Q ! Vdq = ! 0 0 2 q Q dq = C 2C 2 1 U = CVB 2 Energy density 1 Show ! u = 2 ! 0 E Carl Bromberg - Prof. of Physics 2 6 Potential of a point charge Move test charge q’ toward charge q. – ΔU is negative of work done by field in this motion r2 r2 qq% dr !U = " $ Felec # ds = " 4&' 0 $ r 2 r1 r1 qq% = 4&' 0 r2 (1 1+ * r " r - (> 0) ) 2 1, q – Potential of a point charge Lecture 2 r1 dr is negative ds = dr r̂ E= !U !V = = V (r2 ) # V (r1 ) q" V is a scalar ! q# q V (!) = 0 V (r) = 4!" 0 r q 4!"0 r 2 r̂ + dV q E=! r̂ = r̂ 2 dr 4"# 0 r becomes -Grad (V) Carl Bromberg - Prof. of Physics 7 Potential due to a charge distribution Two ways to get potential of a charge distribution – Line integral of a known electric field – Integration of point charge potentials !V = " $ E # ds V (r) = 1 4%& 0 Example: spherical shell with charge density σ – Electric field known from Gauss’s Law (same as point charge r > R, and zero r < R) – Potential obtained from line integral along E !V = " outside point charge V (r) = potential Lecture 2 Q 4#$ 0 Q 4!" 0 r r dr % r2 R = E= dq $r Q 4!" 0 r 2 r̂ Q &1 1 ) " + ( 4#$ 0 ' r R * Q r # R V (r) = 4!" 0 R r < R inside potential is constant Carl Bromberg - Prof. of Physics 8 Potential by integration over point charges dq Potential on axis from ring with charge density λ 2" 1 dq !R V (r) = = d$ 1 % # 4!" 0 r 2 2 2 4"# 0 ( z + R ) 0 1 Q ( 2 2 #2 = z +R ) 4!" 0 != Q 2" R Obtain E on axis from (–)Gradient of V 3 dV Q ( 2 2 !2 E=! k̂ = z z + R ) k̂ dz 4"# 0 Note: z(z + R 2 ) 1 2 !2 = cos" Why V ? More tools available to determine V than E Lecture 2 Carl Bromberg - Prof. of Physics 9 Conductors and static electric fields e– Move electrons from pebble to a metal surface. Pebble becomes charged +q. e– When charges stop moving, the electric field within the conductor is zero, charge is “pulled” to the surface. Also, Gauss’s Law requires that the charge density within this conductor is zero. When charges stop moving, the components of the electric field parallel to the surface, E|| = zero. Also, Gauss’s Law requires that at the surface the electric field normal component, E⊥ = σ /ε0 . The electric potential is a constant throughout the conductor. Later we will learn “method of images” to determine fields & charges Lecture 2 Carl Bromberg - Prof. of Physics 10 Magnetic fields A charge q moves at a velocity v in magnetic field B. – Force on the charge (use right hand rule for + charges) F = qv ! B – Circular motion for v perpendicular to B 2 F = mv r = qvB A current I flows in a thin wire – Force on small segment or on length L dF = I d! ! B F= IL!B – Force between parallel straight wires µ0 I1 I 2 L #7 F= ; µ0 = 4! " 10 T $ m/A 4! d Attractive for same I’s Lecture 2 straight wire B µ0 I ˆ B= " 2! r right hand rule Carl Bromberg - Prof. of Physics 11 Magnetic fields from currents Ampere’s Law – Closed path integral around current I " B ! ds = µ 0 I Choose a circular path! – Example: long straight wire carrying current I # B ! ds = 2"rB = µ I 0 µ0 I B= 2"r µ0 I ^ φ B= 2"r Biot-Savart Law – Magnetic field from small current element dI µ 0 I d! " r̂ dB = 4! r 2 Lecture 2 Carl Bromberg - Prof. of Physics 12