Electrical Potential Energy and Electric Potential (Chapter 29)

Electrical Potential Energy and Electric Potential (Chapter 29)

A Refresher Course on Gravity and Mechanical Energy Total mechanical energy: E mech = K + U, K= 1 2 mv2,u = potential energy f W = F!" ids "! = "K = K f # K i = 1 2 mv 2 f # 1 2 mv i2 i Work done by a force on particle moving from i to f Conservative Forces: W = #"U = #( U f # U ) i For a conservative force, the work done is independent of the path!"!" & %U F = #$ U = # %x i# + %U %y j# + %U ' ( %z k# ) * + "E mech = 0 = "K + "U The total energy is conserved for conservative forces

Conservative Force: Work is Independent of Path The work done by F in going from A to B doesn t depend on the path of the particle. A particle falls in the earth s gravity a height!y.

Calculating the Work: Direction is Everything Maximum work: Force and displacement parallel. Particle is accelerated by force. Zero work: Force and displacement perpendicular. No change in kinetic energy. dw = F!" ids " = F!" cos!ds W = f " i dw Negative work: Force and displacement anti-parallel. The particle loses kinetic energy moving against the force.

!"! GMm F g =! r # r 2 Gravity is a Conservative Force W = W = B "!G Mm r! % # $ r 2 & 'ids " " ( =!G Mm r! % ( # $ r 2 & 'i drr! + d))! A B "!G Mm % " ( # $ r 2 & ' dr = GMm 1! 1 % # $ r B r A & ' A B A ( ) r! = vector pointing in radial direction along line connecting M and m!! = unit vector perpendicular to r! (represents motion along line that doesn't change the distance between masses. It is tangent to the surface of a sphere of constant radius r)

Gravity!" F g = mg " Gravity vs. Electrostatics: Electrostatic Fields Are Also Conservative! g " =! GM r 2 r # (for a point mass M) g " =!mg j # (near surface of earth) Electrostatics!" F E = qe!"!" kq E = r r# (for a point charge Q) 2!"! E = # j (for an infinite sheet of charge) 2" 0 Gravitational forces and fields have the exact same form as electric forces and fields. Therefore work and potential energy for electric fields will also be identical to gravity. f!u g = " F!" gids " # f!v g = " g " ids " i i #!U g = m!v g f!u E = " F!" E ids " # f!v E = " E!" ids " i i #!U E = q!v E

Electrical Potential Energy: Understanding Through Examples

Uniform Constant Electric Field: Just Like Newton s Apple An ideal capacitor has a constant electric field: A charged particle moving between plates behaves the same as a falling object

Uniform Constant Electric Field: Potential Energy of a Test Charge U E = U 0 + qey Note: Direction of increasing potential energy depends on the sign of the charge. Also the value U 0 is irrelevant since it does not affect the work done. Positive Test Charge (q>0) Negative Test Charge (q<0) Increasing U E y y=0 Increasing U E y

The Potential Energy of Point Charges Consider two point charges, q 1 and q 2, separated by a distance r. The electric potential energy is This is explicitly the energy of the system, not the energy of just q 1 or q 2. Note that the potential energy of two charged particles approaches zero as r " #.

How do we get the Coulomb s law from the potential energy?

Colliding Charges & Conservation of Energy

Colliding Like Charges: Repulsion For two charges of the same sign the force is repulsive and potential energy positive. Let us imagine that we fire two charges at each other from far away. Initially they have only kinetic energy: E mech = 1 2 m v 2 1 1 + 1 2 m v 2 2 2 As they approach, the particles slow down and the potential energy increases until E mech = U E (r min ) At r min the particles have 0 velocity and can no longer get any closer without violating energy conservation. They therefore start to move away from each other.

Colliding Charges: Opposites Attract For two charges of the opposite sign the force is attractive and potential energy negative but they can escape each other if E>0 We have two opposite charges sitting next to each other. Initially the total energy E mech = 1 2 m 1v 1 2 + 1 2 m 2v 2 2 + U E (r 0 ) > 0 If the initial velocities are opposite, the particles fly apart by which they slow down and U E increases until it reaches 0 for very large r. However, even at infinite r, the kinetic energies are non-zero and thus they continue to separate.

Colliding Charges: Bound States For two charges of the opposite sign the force is attractive and potential energy negative. This is how atoms are held together, electrons orbiting the nucleus. This is known as a bound state because the particle are bound to each other by their mutual electric attraction. Let us imagine that we have two opposite charges sitting next to each other. Initially the total energy E mech = 1 2 m v 2 1 1 + 1 2 m v 2 2 2 + U E (r 0 ) < 0 E mech is negative. If the initial velocities are opposite, the particles fly apart by which they slow down and U E increases until E mech = U E (r max ) The particles can no longer get further apart. They therefore change direction and start to approach each other again.

U E = r ij = r i Potential Energy of Multiple Charges N j *1 )) j =1! * r! i=1 j # 1 & $ % 4!" 0 ' ( q i q j r ij = ) i< j # 1 & q i q j $ % 4!" 0 ' ( r ij = separation between charges i and j y q 1 q 2 r 2 q 3 U E = # $ % 1 4!" 0 & ' ( # $ % q 1 q 2 r 12 + q 1q 3 r 13 + q 1q 4 r 14 + q 2q 3 r 23 + q 2q 4 r 24 + q 3q 4 r 34 & ' ( r 1 r 3 r 4 q 4 For N charges, there is the potential energy is a sum of N(N-1)/2 terms. x

Electric Potential: Why We Need It. Source Charges! Electric Field Electric Field! Force on Test Charges Electric Potential Energy = Interaction Energy Between Source + Test Charges Potential Energy of Test Charge:!" " E( r ) =!(r " & ) "!" 1 """ ') 4#$ 0 r " r % r ' (!" 2 + "!" dv ' electric field created by 'source' charges ' ( % r ' * + r % r '!"!" F = qe the force on a 'test' charge.,u E = %" F!" ids " Electric Potential of Field: f!u E = "!K = " qe!" ids " f $ # = q "# E!" ids " ' & ) % ( = q!v f!v = "# E!" ids " * V = U E / q i i i Independent of test charges! The electric potential Units: 1 volt= 1V =1 J/C

Does the Electric Potential Offer a Real Advantage Over the Potential Energy? For a single test charge in an electric field of the source charges we have U E =qv For a single test charge both U E and V give us the same information and are just as easy to work with. What about multiple test charges? U E = (interaction with field of source charges)+(interaction between test charges) N U E =! q i V(x i ) + 1 2 i=1! i< j 1 4"# 0 q i q j r ij

The Potential Energy of a Dipole $!! = r! " F "! (general definition of torque) $ 2% W=!! in # d# (work done by torque for rotation about n # ) % 1 n r F $ 1 For an electric dipole:!! = p "! " E "! % 2 $ 2% W = & pe sin# d# = pe cos$ 2 & pe cos$ 1 $ 1 ( ) W = & U 2 & U 1 ' U($) = & pe cos$

Energy of Dipole in Electric Field The potential energy of an electric dipole p in a uniform electric field E is The potential energy is minimum at %= 0 where the dipole is aligned with the electric field.

Energy Conservation and Electric Potential Electric potential V is not an energy! U=qV is an energy. Conservation of energy says: 1 2 mv2 initial + U initial = 1 2 mv2 final + U final OR EQUIVALENTLY: 1 2 mv2 initial + qv initial = 1 2 mv2 final + qv final We will mostly use V instead of U from here on. Like with potential energy, it is the difference that is important 1 2 mv 2 i! 1 2 mv 2 f = q"v = q(v f! V i )

What does the electric potential tell us about the motion of charged particles 1 2 mv 2 f = 1 2 mvi2! q"v # v f = v 2 i! 2q m "V (1)q<0, particle is accelerated in direction in which V increases,!v>0 (2) q>0, particle is accelerated in direction in which V decreases,!v<0

How a positive charge moves in an electric potential: How a negative charge moves in an electric potential: i f - - Negative charge decelerates as it moves through negative potential difference - f!v<0 - i Direction of increasing V!V>0 Negative charge accelerates as it moves through positive potential difference

Electric Field and Potential Inside Ideal Capacitor s! V(s) =!" Eid x! = + " E dx = Es 0 s 0 The electric potential decreases in the direction that E points! The capacitor voltage or voltages across capacitor is the voltage difference between plates:!v C = V + " V " = Ed E =!V C / d # The capacitor voltage immediately gives us the electric field

How to Place a Voltage Across a Capacitor A battery (or DC generator) connected to the plates of a capacitor produces a capacitor voltage equal to the battery voltage. The charge on the plates is determined by the battery voltage and geometry of the capacitor " Q = ±! A 0 % # $ d & ' (V C = C(V C " C =! 0A% # $ d & ' =! 0 plate area plate separation C is known as the capacitance and is property of the geometry.

The Electric Potential of a Point Charge B!V = V B " V A = "# Eid! s! = " kq ˆr r i(drˆr + d$ ˆ$) 2 A B # A B!V = " kq 1 r dr = kq % 1 " 1 ( # 2 & ' r B r A ) * A Only potential differences have any meaning!!! Only changes in potential lead to a force! ASSUME: V(r) = 0 at r =! IT FOLLOWS THAT: V(r) = k Q r = 1 Q 4"# 0 r

What about a uniformly charged sphere? What is the electric potential?

Visualizing the potential landscape of a dipole Positive Charge Negative Charge

Concentric Spherical Shells Two concentric spherical shells First has radius R 1 and charge Q 1 =+Q Second has radius R 2 and charge Q 2 =-Q

Use Gauss s Law for Spherical Shells... R 2!V c = " E!" # i ˆrdr = " 1 R 1 Q # dr = 1 & Q " Q 4$% 0 r 2 4$% 0 ' ( R 1 R 2 R 1 R 2 ) * + = Q & R " R ) 2 1 ' ( 4$% 0 R 1 R 2 * +

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Use Gauss s Law for the Electric Field... a < r < b : E!" =! 2"# 0 r ˆr From Gauss s Law!= charge per unit length b "V c =! 1 % 2#$ 0 r dr =! lnb & lna 2#$ 0 a ( ) =! 2#$ 0 ln( b a )

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 r i is the distance from charge q i 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.

The Electric Potential is Much Easier to Work With Than the E-Field. V is a scalar and the electric field is a vector. For multiple and continuous charge distributions, it is much easier to add the scalar V for each charge element than the vectors E for each element. V(r! ) =! V i (r! ) = 1 q!! i! i 4"# 0 i r $ r i V(r! ) = dv = 1 dq %! 4"# 0 %! = 1! &(r r $ r ' 4"# % ')!! 0 r dv ' $ r ' Keeping track of the direction for E from each charge element dq is usually very difficult. In Ch. 30 we will show how to obtain E in simple manner if V(r) is known.

A Quick Example: The Potential of a Ring of Charge In Ch. 27, we calculated the electric field for this problem on the axis of the ring. Now we will calculate V. dq =!dl =!(ad") r = a 2 + x 2

Potential of Charged Ring (cont.) V = V i! dv = V = + i * * $ % & 1 4"# 0 Q 4"# 0 r = Q 4"# 0 1 a 2 + x 2 ' ( ) dq r = $ % & 1 4"# 0 r ' ( ) * dq