Chapter 23 Electric Potential

Goals for Chapter 23 To calculate electric potential energy of a group of charges To understand electric potential To calculate electric potential due to a collection of charges To use equipotential surfaces to understand electric potential To calculate E field using electric potential

Introduction How is electric potential related to welding? Electric potential energy is an integral part of our technological society. What is the difference between electric potential and electric potential energy? How is electric potential energy related to charge and the electric field?

Electric potential energy in a uniform field Behavior of point charge in uniform electric field is analogous to the motion of a baseball in a uniform gravitational field.

Electric potential energy in a uniform field Behavior of point charge in uniform electric field is analogous to the motion of a baseball in a uniform gravitational field. Ball speeds up g

Electric potential energy in a uniform field Behavior of point charge in uniform electric field is analogous to the motion of a baseball in a uniform gravitational field. + charge speeds up E

A positive charge moving in a uniform field If positive charge moves in direction of field, potential energy decreases (KE increases!)

A positive charge moving in a uniform field If positive charge moves in direction of field, potential energy decreases (KE increases!) If + charge moves opposite field, potential energy increases.

When a positive charge moves in the direction of the electric field, Q23.1 When a positive charge moves in the direction of the electric field, Motion A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases. +q Answer: B

A23.1 When a positive charge moves in the direction of the electric field, Motion A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases. +q

A negative charge moving in a uniform field Positive charge movement in an E field is like normal mass moving in a uniform gravitational field. Move with the field direction, KE increases Move against the field direction, U increases Overall, total energy U + KE is constant. With negative charges, just reverse the sign! 

A negative charge moving in a uniform field If negative charge moves in direction of E field, potential energy increases…

A negative charge moving in a uniform field If negative charge moves in direction of field, potential energy increases, but if - charge moves opposite field, potential energy decreases.

When a negative charge moves in the direction of the electric field, Q23.3 When a negative charge moves in the direction of the electric field, Motion –q A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases. Answer: C

A23.3 When a negative charge moves in the direction of the electric field, Motion –q A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases.

A negative charge moving in a uniform field Positive charge movement in an E field: Move with the field direction, KE increases Move against the field direction, U increases Overall, total energy U + KE is constant. Negative charges movement in an E field: Move with the field direction, KE decreases Move against the field direction, U decreases

Q23.2 When a positive charge moves opposite to the direction of the electric field, Motion +q A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases. Answer: C

A23.2 When a positive charge moves opposite to the direction of the electric field, Motion +q A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases.

Q23.4 When a negative charge moves opposite to the direction of the electric field, Motion –q A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases. Answer: B

A23.4 When a negative charge moves opposite to the direction of the electric field, Motion –q A. the field does positive work on it and the potential energy increases. B. the field does positive work on it and the potential energy decreases. C. the field does negative work on it and the potential energy increases. D. the field does negative work on it and the potential energy decreases.

Electric potential energy of two point charges What is the change in electric potential energy of charge q0 moving along a radial line? b Test charge q0 moves from a to b in the field created by q a + q (fixed)

Electric potential energy of two point charges The electric potential energy of charge q0 moving along a radial line NOTE!! This force is NOT constant! It decreases with distance away from +q ! You HAVE to integrate!

Electric potential energy of two point charges The electric potential energy of charge q0 moving along a radial line NOTE!! This force is NOT constant! It decreases with distance away from +q ! You HAVE to integrate!

Electric potential energy of two point charges The electric potential energy of charge q0 moving along a radial line  

Graphs of the potential energy Electrical Potential Energy Define U = 0 @ r = ∞ Sign depends on charges

Electric potential energy of two point charges Electric potential is same whether q0 moves along an arbitrary path!

Electric potential energy of two point charges Electric potential is same whether q0 moves along an arbitrary path!

Example 23.1 – Conservation of Energy Positron (+ electron) moves near alpha particle (+2e with mass m=6.64 x 10-27 kg) Interacts electrically; at r = 1.00 x 10-10 m moving away at v = 3.00 x 106 m/s. What is positron speed when twice as far? At ∞ ? Assume a particle doesn’t move.

Example 23.1 – Conservation of Energy Positron moves near alpha particle (m=6.64 x 10-27 kg) Interacts electrically; at r = 1.00 x 10-10 m moving away at v = 3.00 x 106 m/s. What is positron speed when twice as far? At ∞ ? Assume a particle doesn’t move.

Example 23.1 – Conservation of Energy Positron moves near alpha particle (m=6.64 x 10-27 kg) Interacts electrically; at r = 1.00 x 10-10 m moving away at v = 3.00 x 106 m/s. What is positron speed when twice as far? At ∞ ? Assume a particle doesn’t move.

Example 23.1 – Conservation of Energy Positron moves near alpha particle (m=6.64 x 10-27 kg) Interacts electrically; at r = 1.00 x 10-10 m moving away at v = 3.00 x 106 m/s. What is positron speed when twice as far? At ∞ ? Assume a particle doesn’t move.  

Problem 23. 1 A point charge with a charge q1 = 3.90μC is held stationary at the origin. A second point charge with a charge q2 = -4.80μC moves from the point x= 0.170m , y=0 to the point x= 0.230m , y= 0.270m . How much work is done by the electric force on q2?

Electrical potential with several point charges Electrical PE associated with q0 depends on other charges and distances from q0 Key example 23.2: Two point charges on x axis q1 = -e at x = 0 q2 = +e at x = a Work done to bring q3 = +e from ∞ to x = 2a?

Potential from Superposition of Charges

D. not enough information given to decide Q23.5 The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is Charge #2 +q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide Answer: B

A23.5 The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is Charge #2 +q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide

D. not enough information given to decide Q23.6 The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is Charge #2 –q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide Answer: B

A23.6 The electric potential energy of two point charges approaches zero as the two point charges move farther away from each other. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential energy of the system of three charges is Charge #2 –q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide

D. not enough information given to decide Q23.7 The electric potential due to a point charge approaches zero as you move farther away from the charge. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is Charge #2 +q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide Answer: A

A23.7 The electric potential due to a point charge approaches zero as you move farther away from the charge. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is Charge #2 +q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide

D. not enough information given to decide Q23.8 The electric potential due to a point charge approaches zero as you move farther away from the charge. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is Charge #2 –q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide Answer: B

A23.8 The electric potential due to a point charge approaches zero as you move farther away from the charge. If the three point charges shown here lie at the vertices of an equilateral triangle, the electric potential at the center of the triangle is Charge #2 –q Charge #1 +q y –q x Charge #3 positive. negative. C. zero. D. not enough information given to decide

Problem 23. 8 Three equal 1.70-μC point charges are placed at the corners of an equilateral triangle whose sides are 0.600m long. What is the potential energy of the system? (Take as zero the potential energy of the three charges when they are infinitely far apart.)

Electric potential Potential is potential energy per unit charge. Think of potential difference between points a and b in either of two ways.

Electric potential Potential is potential energy per unit charge. One Way: Potential of a with respect to b (Vab = Va – Vb) work done by the electric force on a unit charge that moves from a to b in the field.   if b is at a lower potential than a, work done moving + charge is positive (it speeds up!)

Electric potential Potential is potential energy per unit charge. Potential of a with respect to b (Vab = Va – Vb) equals: work done by the electric force on a unit charge that moves from a to b in the field. Think of gravity doing positive work on a falling mass   Note limits of integration, and their order!

Electric potential analogy to gravity… Fg and dl vectors are in the same direction Work is positive Potential decreases Va > Vb dl g b

Electric potential analogy to gravity… Fe and dl vectors are in the same direction Work is positive Potential decreases Va > Vb + dl E b

Electric potential Potential is potential energy per unit charge. Another Way: Potential of a with respect to b (Vab = Va – Vb) work done by you to move a unit charge slowly the other way, from b to a against the electric force. Think of lifting a mass against gravitational force!

Electric potential analogy to gravity… Fg and dl vectors are in opposite directions Work is negative Potential increases Va > Vb dl g b

Electric potential Potential is potential energy per unit charge. Think of potential difference between points a and b in either of two ways. Potential of a with respect to b (Vab = Va – Vb) equals: work done by the electric force when a unit charge moves from a to b. work done by you to move a unit charge slowly from b to a against the electric force.

Potential due to charges Potential is a scalar Three “standard” forms

Finding electric potential from the electric field Move in direction of E field electric potential decreases, but if you move opposite the field, the potential increases. For EITHER + or – charge!

Potential due to two point charges Example 23.3: Proton moves .50 m in a straight line between a and b. E = 1.5x107 V/m from a to b. Force on proton? Work done by field? Va – Vb?

Problem 23. 13 A small particle has charge -3.10μC and mass 2.40×10−4kg . It moves from point A, where the electric potential is VA = 300V , to point B, where the electric potential VB = 990V is greater than the potential at point A. The electric force is the only force acting on the particle. The particle has a speed of 4.90m/s at point A. What is its speed at point B?

Potential due to two point charges Example 23.4: Dipole with q1 = +12 nC & q2 = -12 nC d = 10 cm Va ? Vb ? Vc ?

Finding potential by integration Example 23.6: Potential at a distance r from a point charge? Here a = r, b = 

Finding potential by integration Example 23.6: Potential at a distance r from a point charge? Or equally Here a = r, b = 

Moving through a potential difference Example 23.7 combines electric potential with energy conservation. Dust particle m = 5.0 x 10-9 kg, charge +2.0 nC Starts at rest. Moves from a to b. Speed at b?

Moving through a potential difference Field is NOT constant! Forces are NOT constant!! Use Energy Conservation instead… Ka+Ua= Kb+Ub where U = q0V & V = Skq/r

Calculating electric potential Example 23.8 (a charged conducting sphere). Total charge q in sphere of radius R V = ? everywhere

Calculating electric potential Example 23.8 (a charged conducting sphere). Total charge q in sphere of radius R Start with E field! E outside R looks like pt charge! E inside = 0 (conductor!

Calculating electric potential Example 23.8 (a charged conducting sphere). Total charge q in sphere of radius R V = ? Everywhere V inside must be? NOT zero! Just CONSTANT!! V outside = point charge!

Oppositely charged parallel plates Example 23.9: Find potential at any height y between 2 oppositely charged plates

Example 23.9: Oppositely charged parallel plates

Example 23.9: Oppositely charged parallel plates

An infinite line charge or conducting cylinder Example 23.10 – Potential at r away from line?

An infinite line charge or conducting cylinder Example 23.10 – Potential at r away from line?

An infinite line charge or conducting cylinder Example 23.10 – Potential at r away from line?

An infinite line charge or conducting cylinder Example 23.10 – Potential at r away from line?

An infinite line charge or conducting cylinder Example 23.10 – Potential at r away from line? If Vb = 0 at infinite distance, won’t Va go to infinity at r = 0?? Yes! So define V= 0 elsewhere

An infinite line charge or conducting cylinder Consider cylinder… Same result for r > R What about inside conducting cylinder??

A ring of charge Charge Q uniformly around thin ring of radius a. VP=?

A finite line of charge Example 23.12 – Potential at P a distance x from ring?

Equipotential surfaces and field lines Equipotential surface is surface on which electric potential is same at every point. Field lines & equipotential surfaces are always mutually perpendicular.

Equipotential surfaces and field lines Equipotential surface is surface on which electric potential is same at every point. Field lines & equipotential surfaces are always mutually perpendicular.

Equipotential surfaces and field lines Equipotential surface is surface on which electric potential is same at every point. Field lines & equipotential surfaces are always mutually perpendicular.

Equipotentials and conductors When all charges are at rest: Surface of conductor is always an equipotential surface. E field just outside conductor is always perpendicular to surface Entire solid volume of conductor is at same potential.

Potential gradient Work/unit charge Consider a uniform vector E field Consider moving a unit charge along x-axis a very small distance from x to x+Δx (at constant y and z) Work done against field from x to x+Δx: Work/unit charge Work against field by you

Potential gradient Consider a uniform vector E field Consider moving a unit charge along x-axis a very small distance from x to x+Δx (at constant y and z) in this field YOU have to do work against field from x to x+Δx Remember Work by you against field = DPE You lift mass up against gravity = gain in grav. PE! You push + charge against E field = gain in elec. PE! Your work against the field increases the potential energy.

Potential gradient Consider a uniform vector E field Consider moving a unit charge along x-axis a small distance from x to x+Δx (at constant y and z) But Work (per unit charge) done against field = D V Initial position Final position

Potential gradient Consider a uniform vector E field Consider moving a unit charge along x-axis a small distance from x to x+Δx (at constant y and z) But Work (per unit charge) done against field = D V! In the limit as Δx goes to 0…

Potential gradient So uniform vector E field is the gradient of a scalar Potential

Potential gradient Create E as the gradient of the Potential

Potential gradient Create E as the gradient of the Potential

Potential gradient Analogy to Gravitational Potential Gradient = direction of force along steepest slope Lines of constant height = constant “gh” = (Potential Energy/unit mass) Top down view

Potential gradient Analogy to Gravitational Potential Gradient = direction of force along steepest slope Lines of equal gravitational potential (regardless of mass) Top down view Side view

Potential gradient Analogy to Gravitational Potential Gradient = direction of force along steepest slope Biggest change in (m)gh for small distance sideways Top down view (m)gh decreases in y direction quickly with small movement in x

Potential gradient Analogy to Gravitational Potential Gradient = direction of force along steepest slope Steepest slope means largest net force in direction of gravitational field Top down view Largest negative change in (m)gh with small change in horizontal direction

Largest negative change in (m)gh with small change in xy direction Potential gradient Analogy to Gravitational Potential Gradient = direction of force along steepest slope Steepest slope means largest net force in direction of gravitational field Top down view Largest negative change in (m)gh with small change in xy direction

Potential gradient Create E as the gradient of the Potential 3V Lines of constant electric potential (Volts) (Potential Energy/unit charge) 4V 5V 6V

Potential gradient Create E as the gradient of the Potential 3V Lines of constant electric potential (Volts) (Potential Energy/unit charge) 4V 5V 6V Largest negative change in Volts with small change in horizontal direction = direction of E field!

Potential gradient Create E as the gradient of the Potential Potential decreases in direction of + E field Va – Vb is positive if E field points from a to b “Grad V” is from final position – initial = Vb – Va <0 So is positive!