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Unformatted text preview: Electrostatic Potential and Energy
Chapter Outline ENGINEERING PHYSICS II
PHY 303L •The electrostatic potential
•Calculation of the Potential from the Field Chapter 25: Electrostatic Potential and Energy •Potential in Conductors
•Calculation of the Field from the Potential Maxim Tsoi •Energy of Systems of Charges
Physics Department,
The University of Texas at Austin http://www.ph.utexas.edu/~tsoi
303L: Electric Potential (Ch.25) 303L: Electric Potential (Ch.25) Potential difference and electric potential Potential difference and electric potential potential energy of the chargefield system • When a charge is displaced by ds, the work done by the electric field rr
rr
F ⋅ ds = q0 E ⋅ ds is: • As work is done by the field, the potential energy of the chargefield
system is changed by an amount: rr
dU = − q0 E ⋅ ds
• For a finite displacement of the charge from A to B the change in
potential energy is: electric potential • Electric potential is a scalar characteristic of an V= electric field, independent of any charges that may be
placed in the field
• Potential difference take the value of the electric potential to be zero at some convenient point ∆V ≡ • The work done by an external agent to move an electric
charge q through the field at constant velocity is
• The SI unit of both electric potential and potential Br
r
∆U = − q0 ∫A E ⋅ ds • The electric force is conservative the line integral does not depend on the path taken from A to B 303L: Electric Potential (Ch.25) U
q0 Br
r
∆U
= − ∫A E ⋅ ds
q0 W = q∆ V
1 V = 1 J/C difference is volt (V)
• Electric field can also be expressed in V/m
• Energy can be measured in eV 1 N/C = 1 V/m the energy a chargefield system gains or loses when a charge of magnitude e is moved through a potential difference of 1V 303L: Electric Potential (Ch.25) Potential difference Potential difference in a uniform electric field equipotential surfaces • Potential difference between two points A and B separated
by a distance s=d (sE)
B VB − V A = ∆V = − ∫
A • Consider a charged particle moving between two
points A and B so that s is not parallel to E rr
B
E ⋅ ds = − ∫ Eds = − Ed
A • Electric field lines always point in the direction of
decreasing electric potential rBr
rr
Br
r
VB − V A = ∆V = − ∫ E ⋅ ds = − E ∫ ds = − E ⋅ s
A
A
• The change in potential energy of the chargefield
system is rr
∆U = q0 ∆V = − q0 E ⋅ s • Positive charge and field system looses potential energy
when the charge moves in the direction of the field: ∆U = q0 ∆V = − q0 Ed
• Negative charge and field system gains potential energy • VBVA=VCVA VB=VC • Equipotential surface surface consisting of points having the same electric potential when the charge moves in the direction of the field 303L: Electric Potential (Ch.25) 303L: Electric Potential (Ch.25) 1 Electric Potential and Potential Energy Electric Potential and Potential Energy due to point charges due to two or more point charges • Potential difference between two points A and B is • by applying the superposition principle Br
B
r
r
q
ˆ
VB − V A = − ∫ E ⋅ ds = − ∫ ke 2 r ⋅ ds
A
A
r The total electric potential at some point P due to several point
charges is the sum of the potentials due to the individual charges r
ˆ
or since r ⋅ ds = ds cos θ = dr VB − V A = − k e q ∫ rB rA • With reference choice V=0 at r = ∞, the electric potential created by point
charge is: r dr ke q B ke q ke q
=
−
=
r2
r rA
rB
rA V = ke V = ke ∑
i qi
ri q
r 303L: Electric Potential (Ch.25) 303L: Electric Potential (Ch.25) Electric Potential and Potential Energy Electric Field from Electric Potential due to point charges due to point charges • V1 is the electric potential at a point P due to charge q1 • electric field and electric potential are related via
Br
r
∆V = − ∫A E ⋅ ds • To bring a second charge q2 from infinity to P without • The potential difference between two points a distance acceleration, an external agent must do the work ds apart is rr
dV = − E ⋅ ds W = q 2V1
• This work is transferred to the system and appears as
Er = − potential energy U = q2V1 = ke q1q2
r12 or if the electric field has only x component dV
dr dV = − E x dx ⇒ Ex = − dV
dx • Electric field is a measure of the rate of change • If the system consists of more than two (e.g., three) with position of the electric potential charges: qq qq q q U = ke 1 2 + 1 3 + 2 3 r
r13
r23 12 Ex = − Example 2 303L: Electric Potential (Ch.25) dV
dx Ey = − dV
dy Ez = − dV
dz 303L: Electric Potential (Ch.25) Electric Potential Electric Potential due to continuous charge distribution due to a charged conductor • electric potential at some point P due to the charge • Every point on the surface element dq is of a charged conductor in dV = ke dq
r equilibrium is at the same
electric potential
Br
r
VB − V A = − ∫ E ⋅ ds
A • The total potential at point P is V = ke ∫ dq
r • The potential is the conductor and equal to • Reference: the electric potential is taken to be zero its value at the surface when point P is infinitely far from charge distribution 303L: Electric Potential (Ch.25) electric constant everywhere inside Examples 36 303L: Electric Potential (Ch.25) Example 7 2 SUMMARY
Electrostatic Potential and Energy
• When a test charge q0 is moved between points A and
B in an electric field E, the change in the potential
energy of the chargefield system is
• The electric potential: V= • The potential difference between points A and B:
in a uniform field: Br
r
∆U = − q0 ∫A E ⋅ ds U
q0 ∆V ≡ ∆V = − Ed • An equipotential surface is one on which all points are at the same electric potential • The potential energy associated with a pair of point U = ke charges:
• The negative derivative of the electric potential gives
the components of the electric field:
• The electric potential due to a continuous charge
distribution: Br
r
∆U
= − ∫A E ⋅ d s
q0 Ex = − dV
dx q1q2
r12 Ey = − dV
dy V = ke ∫ Ez = − dV
dz dq
r 303L: Electric Potential (Ch.25) 3 ...
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This note was uploaded on 10/20/2009 for the course PHY 59090 taught by Professor Tsoi during the Spring '09 term at University of Texas.
 Spring '09
 TSOI
 Energy

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