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Review Electric Field Chapter 29. The Electric Potential At any time, millions of light bulbs are transforming electric energy into light and thermal energy. Just as electric fields allowed us to understand electric forces, Electric Potential allows us to understand electric energy. Chapter Goal: To calculate and use the electric potential and electric potential energy. Stop to think 29.1 page 885 Stop to think 29.2 page 889 Stop to think 29.3 page 893 Stop to think 29.4 page 896 Stop to think 29.5 page 898 Electrical Potential Energy When a test charge is placed in an electric field, it experiences a force E F qE q s If s is an infinitesimal displacement of test charge, then the work done by electric force during the motion of the charge is given by F s qE s 4 Uelec Uo Eqs Uelec Welec where s is measured from the negative plate and U0 is the potential energy at the negative plate (s = 0). It will often be convenient to choose U0 = 0, but the choice has no physical consequences because it doesn’t affect ΔUelec, the change in the electric potential energy. Only the change is significant. Ugrav Uo mgy Electrical Potential Energy F qE α q A ds s E ds B B U UB UA q E ds A ds is oriented tangent to a path through space For all paths: UB UA qsE cos α The electric force is conservative 7 The potential energy of two point charges xf Uf Ui Welec F 1on2 dx xi xy Kq1q 2 1 xf Kq1q 2 Kq1q 2 dx Kq1q 2( ) | xi 2 x xf xi x xi The Potential Energy of Point Charges P886. Consider two point charges, q1 and q2, separated by a distance r. The electric potential energy is This is explicitly the energy of the system, not the energy of just q1 or q2. Note that the potential energy of two charged particles approaches zero as r . The Potential Energy of a Dipole P889. The potential energy of an electric dipole p in a uniform electric field E is The potential energy is minimum at ø = 0° where the dipole is aligned with the electric field. pE sin The Electric Potential We define the electric potential V (or, for brevity, just the potential) as Charge q is used as a probe to determine the electric potential, but the value of V is independent of q. The electric potential, like the electric field, is a property of the source charges. The unit of electric potential is the joule per coulomb, which is called the volt V: Electric Potential in constant electric field Electric potential is the potential energy per unit charge, U V q The potential is independent of the value of q. The potential has a value at every point in an electric field Only the difference in potential is the meaningful quantity. UB UA sF VB VA sE sE cos α q q q F qE α q A s B E 13 The Electric Potential Inside a ParallelPlate Capacitor The electric potential inside a parallel-plate capacitor is where s is the distance from the negative electrode. The electric potential, like the electric field, exists at all points inside the capacitor. The electric potential is created by the source charges on the capacitor plates and exists whether or not charge q is inside the capacitor. EXAMPLE 29.7 A proton in a capacitor QUESTIONS: EXAMPLE 29.7 A proton in a capacitor EXAMPLE 29.7 A proton in a capacitor EXAMPLE 29.7 A proton in a capacitor Electric Potential for a Point Charge V 0 Point Charge V 0 Q Er k e 2 r ds B E r Q0 B r r r dr 1 Q VB VA Eds Er dr keQ 2 keQ ke r r r 19 The Electric Potential of a Point Charge Let q be the source charge, and let a second charge q', a distance r away, probe the electric potential of q. The potential energy of the two point charges is By definition, the electric potential of charge q is The potential extends through all of space, showing the influence of charge q, but it weakens with distance as 1/r. This expression for V assumes that we have chosen V = 0 to be at r = . Electric Potential: Example Point Charge V 0 equipotential lines Q Vr ke r 21 The Electric Potential of a Charged Sphere In practice, you are more likely to work with a charged sphere, of radius R and total charge Q, than with a point charge. Outside a uniformly charged sphere, the electric potential is identical to that of a point charge Q at the center. That is, Or, in a more useful form, the potential outside a sphere that is charged to potential V0 is The Electric Potential of Many Charges The electric potential V at a point in space is the sum of the potentials due to each charge: where ri is the distance from charge qi to the point in space where the potential is being calculated. In other words, the electric potential, like the electric field, obeys the principle of superposition. EXAMPLE 29.10 The potential of two charges EXAMPLE 29.10 The potential of two charges Potential and Potential Energy • If we know potential then the potential energy of point charge q is U qV (this is similar to the relation between electric force and electric field) F qE 26 Potential Energy: Example Potential energy of two point charges: q Q U ke r U qQ r U σQ 0 σQ 0 attraction repulsion 0 r 0 r 27 Potential Energy: Example Find potential energy of three point charges: q2 q1 r12 r13 r23 U12 ke U U12 U13 U23 q1q2 r12 U13 ke q1q3 r13 U23 ke q2q3 r23 q3 U U12 U13 U23 ke qq qq q1q2 ke 1 3 ke 2 3 r12 r13 r23 28 Electric Potential of Continuous Charge Distribution • Consider a small charge element dq – Treat it as a point charge • The potential at some point due to this charge element is dq dV ke r • To find the total potential, you need to integrate to include the contributions from all the elements V 0 dq V ke r The potential is a scalar sum. The electric field is a vector sum.29 The potential of a ring of charge ri R z 2 2 1 dq 1 dq 1 V Vi 4 0 ri 4 0 R2 z 2 4 0 i 1 N Q R2 z 2 Spherically Symmetric Charge Distribution V 0 V 0 r a Er ke ds Q r a3 r Q Er k e 2 r E C B Q0 B a r r a VB VA Eds Er dr Er dr Er dr a 2 Q dr Q Q Q r 2 2 ke 3 rdr keQ 2 ke 3 (a r ) ke ke 3 2 a r r 2a a 2a a 31 r Spherically Symmetric Charge Distribution Vr ke Q r Q r2 VB ke 3 2 2a a r a r a 32 quiz1 The positive charge is the end view of a positively charged glass rod. A negatively charged particle moves in a circular arc around the glass rod. Is the work done on the charged particle by the rod’s electric field positive, negative or zero? A. Positive B. Negative C. Zero C Quiz 2 A proton is released from rest at point B, where the potential is 0 V. Afterward, the proton A. moves toward A with a steady speed. B. moves toward A with an increasing speed. C. moves toward C with a steady speed. D. moves toward C with an increasing speed. E. remains at rest at B. B Quiz 3 Rank in order, from largest to smallest, the potentials Va to Ve at the points a to e. A. B. C. D. E. Vd = Ve > Vc > Va = Vb Vb = Vc = Ve > Va = Vd Va = Vb = Vc = Vd = Ve Va = Vb > Vc > Vd = Ve Va = Vb = Vd = Ve > Vc D Quiz 4 Rank in order, from largest to smallest, the potential differences ∆V12, ∆V13, and ∆V23 between points 1 and 2, points 1 and 3, and points 2 and 3. A. B. C. D. E. ∆V13 > ∆V12 > ∆V23 ∆V13 = ∆V23 > ∆V12 ∆V13 > ∆V23 > ∆V12 ∆V12 > ∆V13 = ∆V23 ∆V23 > ∆V12 > ∆V13 B