PHY2049: Chapter 2422ConcepTest: Electric Potential ÎWhich point has the largest potential when Q > 0?ÎWhich two points have the same potential? (a) A and C (b) B and E (c) B and D (d) C and D (e) no pairACBDEQESmallest radiusSame radiusPHY2049: Chapter 2423Multiple Charges: SuperpositionÎ3 charges: Find total potential at a point in space− Q3+Q1+Q2xr1r3r231212312 3totQQQVVVVk k krr r−=++= + + V is a scalar¾ No directions to worry about!¾ But you do have to watch signs!+Q1+Q2−Q3PHY2049: Chapter 2424ConcepTest: Electric Potential ÎWhat is V at point A? (a) V > 0 (b) V = 0 (c) V < 0ÎWhat is V at point B? (a) V > 0 (b) V = 0 (c) V < 0Closer to + chargeEqual distance to both chargesABQ2= +50μCQ1= −50μC30 cm26 cm26 cm40 cm60 cmPHY2049: Chapter 2425ConcepTest: Electric Potential ÎWhich configuration gives V = 0 at all points on x axis? (a) A (b) B (c) C (d) All of the above (e) None of the aboveAx+2μC-2μC+1μC-1μCBx+2μC-1μC+1μC-2μCCx+2μC-1μC-2μC+1μCAll points on x axis equidistantfrom each pair of chargesPHY2049: Chapter 2426ConcepTest: Electric Potential ÎAt which point does V = 0? (a) C (b) A (c) D (d) B (e) all of the aboveACBD+Q−QAll points equidistantfrom chargesPHY2049: Chapter 2427ConcepTest: Fields & Potentials ÎFind E and V at the center of the square. (a) E = 0 V = 0 (b) E = 0 V ≠ 0 (c) E ≠ 0V ≠ 0 (d) E ≠ 0V = 0 (e) E = V regardless of the value-Q-Q+Q+QPHY2049: Chapter 2428ConcepTest: Electric Potential ÎYou move a positive charge Q from A to B along the path shown. What is the sign of the work done by you? (a) WAB< 0 (b) WAB= 0 (c) WAB> 0ABNo change in potential sincedistance from center is the samePHY2049: Chapter 2429Calculating E From Electric Potential VÎChange in potential from small change in distance,yxzVEy∂=−∂xyzdV d E dx E dyEdz=− ⋅ =− − −Es,xyzVEx∂=−∂,zxyVEz∂=−∂“Partial derivative wrt x” ⇒ y,z const“Partial derivative wrt y” ⇒ x,z const“Partial derivative wrt z” ⇒ x,y constPHY2049: Chapter 2430Example: V = 30 – 5xÎEquipotential Surfaces: Planes @ constant xV = 30 25 20 15 10x = 0 1 2 3 430 5Vx=−PHY2049: Chapter 2431Find E Field for V = 30 – 5xV = 30 25 20 15 10x = 0 1 2 3 4Differentiate V to get Ex, Ey, Ez30 5500xyzVxVExVEyVEz=−∂=− =∂∂=− =∂∂=− =∂So E = (5,0,0)PHY2049: Chapter 2432Example: Electric Field of Point ChargeÎPoint charge22ˆ,,kq x y z kqErrrrrr⎛⎞=≡⎜⎟⎝⎠G()()1/2 3/2 3222 22233likewise xyzkq kqx kqxExrxyz xyzkqy kqzEErr∂=− = =∂++ ++==222rxyz=++Coulomb’s law, as expectedkqVr=PHY2049: Chapter 2433Potential of Charge DistributionÎGeneralize superposition to continuous distributionÎDistribution can be any shape Line, surface, volumeÎExpress dq in terms of charge density Line or arc: dq = λds or dq = λdx (λ = linear charge density) Surface: dq = σdA (σ = surface charge density) Volume: dq = ρdV (ρ = volume charge density)ÎExpress r in terms of a problem’s “natural” coordinates x, θ, r, …totkdqVr=∫PHY2049: Chapter 2434Example: Charged RingÎFind V at a point z above axis of charged ring of radius R()2022kRdkdqVrzRπλθ==+∫∫2QRλπ=22 222 kR kQVzRzRπλ==++z()3/222zVkQzEzzR∂=− =∂+22dq RdrzRλθ==+Rr2For zkQzREz=PHY2049: Chapter 2435Example: Charged LineÎFind V above midpoint of line of charge Q, length L()/2/222LLkdxkdqVryxλ−==+∫∫/QLλ=2222/4 /2ln/4 /2yL LkyL Lλ⎛⎞++⎜⎟=⎜⎟+−⎝⎠yPxLdq dxλ=22rxy=+PHY2049: Chapter 2436Charged Line: Limit of L yÎRationalize expression inside ln()ÎFor L y22lnLy⎛⎞→⎜⎟⎜⎟⎝⎠()22222222/4 /2/4 /2ln ln/4 /2yL LyL LyyL L⎡⎤++⎛⎞⎢⎥++⎜⎟=⎢⎥⎜⎟⎢⎥+−⎝⎠⎢⎥⎣⎦2lnLVkyλ⎛⎞=⎜⎟⎝⎠PHY2049: Chapter 2437Charged Line (cont)ÎCalculate y component of electric field at midpoint2222/4 /2ln/4 /2yL LVkyL Lλ⎛⎞++⎜⎟=⎜⎟+−⎝⎠22/4yVkLEyyy Lλ∂=− =∂+Agrees with calculationin chapter 22PHY2049: Chapter 2438Example: Charged DiskÎFind V at a point z above axis of charged disk of radius R()0222RkdkdqVrzσπρ ρρ==+∫∫()222VkzRzπσ=+−2(surface charge density)QRσπ=()2dq dA dσσπρρ==zRrρPHY2049: Chapter 2439Charged Disk (cont)ÎCalculate z component of electric fieldÎWhen z very small()222VkzRzπσ=+−2221zVzEkzzRπσ⎛⎞∂=− = −⎜⎟⎜⎟∂+⎝⎠022zEkσπσε=Just like sheet of chargeusing Gauss’ lawPHY2049: Chapter 2440DipoleLike chargesWhere are equilibrium points?Vx−Q+QVx+Q+Q12kQ kQVrr=− +12kQ kQVrr=+No equilibrium since E is never 0Equilibrium is at x = 0, since E = 0E is – dV/dxPHY2049: Chapter 2441Conductors are EquipotentialsÎNo work to move along conductor W = 0 = −qΔVAB⇒ V is constant in conductorÎBut E = 0 inside surface bounded by conductor VC= VA⇒ V is constant within enclosed volumeACBPHY2049: Chapter 2442Conductors in Electrostatic EquilibriumÎElectric field is zero everywhere inside the conductor if E ≠ 0, then charges would move – no equilibrium!!ÎExcess charge on isolated conductor is only on surface Mutual repulsion pushes the charges apartÎElectric field is perpendicular to the surface of a conductor If a parallel component existed, charges would move!!ÎFor irregular shaped conductors, charge density is highest near sharp points, i.e. the field strength is greater therePHY2049: Chapter 2443Spherical Shell+Q----------++++++++++¾ What is charge on inner shell?−Q¾ What is charge on outer shell?+Q¾ What is V vs radius?Constant from 0 < r < r2Falls as kQ / r for r > r2Inner radius = r1Outer radius = r2PHY2049: Chapter 2444ÎA positively charged rod is held near a neutral conducting sphere. A positively charged particle is moved from A to B (A, B both on sphere).ÎThe mechanical work required to cause this motion is (a) positive (b) zero (c) negative (d) depends on the path taken from A to B (e) cannot be determined without more informationConcepTest: Electric EnergyAll points on sphere are at same potentialPHY2049: Chapter 2445ConcepTest: Electric EnergyÎA positively charged rod is held near a neutral conducting sphere. A positively charged particle is moved from A to B (A is on sphere). ÎThe mechanical work required to cause this motion is (a) positive (b) zero (c) negative (d) depends on the path
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