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SPH4UI Electric Potential Mr. Burns

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Electricity has Energy To separate negative and positive charges from each other, work must be done against the force of attraction. Therefore seperated charges are in a higher- energy state. When the charges are brought back together again, energy must be released. It may be in the form of a spark. Consider a small test charge in the electric field around a positive charged object. The field is pushing it in the direction of the field lines. If some external force pushes it the other way, against the direction of the field, work is being done on it. Therefore, a positive charge has its greatest electric potential energy at the upper end (closest to charged object) of the field. Since the field pushes a negative charge the other way, its potential energy is greatest at the lower end. F

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Review: Gravitational Force is conservative Consider a comet in a highly elliptical orbit At point 1, particle has a lot of potential energy, but little kinetic energy Total energy = K + U is constant! At point 2, particle has little potential energy, but a lot of kinetic energy More potential energy Less potential energy pt 1 pt 2 0 U(r)U(r) U(r1)U(r1) U(r2)U(r2)

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Electric Potential Energy We see that potential energy increases as opposite charges recede, and that potential energy increases as like charges approach.

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Electric Potential Energy Electric Potential Energy: Doesn’t it kinda look familiar?

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Electric Potential Energy The electrostatic force is a conservative (=“path independent”) force It is possible to define an electrical potential energy function with this force Work done by a conservative force is equal to the negative of the change in potential energy. Think of the work done by gravity to raise an object to a higher location, since gravity points down and the displacement is up, therefore the work is negative. Yet, the potential energy increases.

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Consider a charged particle traveling through a region of static electric field: A negative charge is attracted to the fixed positive charge negative charge has more potential energy and less kinetic energy far from the fixed positive charge, and… more kinetic energy and less potential energy near the fixed positive charge. But, the total energy is conserved + - We will now discuss electric potential energy and the electrostatic potential…. Electric Force is conservative - - + + - -

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Electric Potential Consider the electric potential energy not just of any charge q 2, but of a unit positive test charge when in the field of any other charge q 1. We call this value of potential energy per unit positive charge the electric potential. The units of electric potential are joules per coulomb, or volts Electric Potential Electric Potential describes how much work is done per unit charge. Recall: Electric Potential Energy:

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Electric Potential Difference The amount of work required per unit charge to move a positive charge from one point to another in the presence of an electric field. The units are also volts + + High Potential Low Potential

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Energy and Charge Movements A positive charge gains electrical potential energy when it is moved in a direction opposite the electric field If a charge is released in the electric field, it experiences a force and accelerates, gaining kinetic energy As it gains kinetic energy, it loses an equal amount of electrical potential energy A negative charge loses electrical potential energy when it moves in the direction opposite the electric field

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Energy and Charge Movements When the electric field is directed downward, point B is at a lower potential than point A A positive test charge that moves from A to B loses electric potential energy It will gain the same amount of kinetic energy as it loses potential energy F=

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Summary of Positive Charge Movements and Energy When a positive charge is placed in an electric field It moves in the direction of the field It moves from a point of higher potential to a point of lower potential Its electrical potential energy decreases Its kinetic energy increases

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Summary of Negative Charge Movements and Energy When a negative charge is placed in an electric field It moves opposite to the direction of the field It moves from a point of lower potential to a point of higher potential It moves from a point where it has higher potential energy to a point where it has lower potential energy. Its electrical potential energy decreases Its kinetic energy increases

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Review of Potential and Potential Energy + + + + Force: - Moves from High Potential (V) to Low Potential (V) - Potential energy (U) decreases - Electric Field (E) does positive work + + - - Force: - Moves from Low Potential (V) to High Potential (V) - Potential energy (U) decreases - Electric Field (E) does positive work - - + + Force: - Moves from High Potential (V) to Low Potential (V) - Potential energy (U) decreases - Electric Field (E) does positive work - - - - Force: - Moves from Low Potential (V) to High Potential (V) - Potential energy (U) decreases - Electric Field (E) does positive work

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Electric potential energy Imagine two positive charges, one with charge Q 1, the other with charge Q 2 : Initially the charges are very far apart, so we say that the initial energy U i of interaction is zero (we are free to define the energy zero somewhere). If we want to push the particles together, this will require work (since they want to repel). the final energy U f of the system will increase by the same amount: U = U f – U i = W Q1Q1 Q1Q1 Q2Q2 Q2Q2 Q1Q1 Q1Q1 Q2Q2 Q2Q2

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Electric potential Energy Pretend q 1 is fixed at the origin. What is the work required to move q 2, initially at infinity, to a distance r away? What if q 2 were negative (but q 1 still positive)? Then the work “required” by us would be negative the charges would like to come together. In this case the final energy is negative! Particles will move to minimize the final potential energy. Remember – work is force times distance: Q1Q1 Q1Q1 Q2Q2 Q2Q2 r

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Electric Potential Energy In addition to discussing the energy of a “test” charge in a Coulomb field, we can speak of the electric potential energy of the field itself! Reasons? – Work had to be done to assemble the charges (from infinity) into their final positions. – This work is the potential energy of the field. – The potential energy of a system of N charges is defined to be the algebraic sum of the potential energy for every pair of charges. This theme continues in the course with E in capacitors and B in inductors – these devices store electric and magnetic energy. We will start with a couple of example calculations…

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Electric Potential Energy Example 1: What is the potential energy of this collection of charges? Step 1:Bring in +2 q from infinity. This costs nothing. -q-q -q-q +2 q d d Step 2:Bring in one -q charge. The force is attractive! The work required is negative: Step 3:Bring in 2nd -q charge. It is attracted to the +2 q, but repelled from the other -q charge. The total work (all 3 charges) is A negative amount of work was required to bring these charges from infinity to where they are now (i.e., the attractive forces between the charges are larger than the repulsive ones).

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ACT 1 Consider the 3 collections of point charges shown below. –Which collection has the smallest potential energy? (a) (b) (c) d d d -Q-Q -Q-Q -Q-Q d d d -Q-Q +Q+Q +Q+Q d d -Q-Q +Q+Q +Q+Q

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ACT 1 Consider the 3 collections of point charges shown below. Which collection has the smallest potential energy? (a) (b) (c) d d d -Q-Q -Q-Q -Q-Q d d d -Q-Q +Q+Q +Q+Q d d -Q-Q +Q+Q +Q+Q We have to do positive work to assemble the charges in (a) since they all have the same charge and will naturally repel each other. In (b) and (c), it’s not clear whether we have to do positive or negative work since there are 2 attractive pairs and one repulsive pair. (a) (b) (c) U 0 (a) (b) (c)

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A If a third charge is added to the system and placed at point A, how does the electric potential energy of the charge collection change ? Two charges which are equal in magnitude, but opposite in sign are placed at equal distances from point A. a) increases b) decreases c) doesn’t change Electric Potential

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Electric potential The potential energy of an added test charge q 0 at point P is just Q1Q1 Q2Q2 Q3Q3 q0q0 r 1p r 2p r 3p Consider that we have three charges fixed in space. Note that this factors: q 0 x (the effects of all other charges) Just as we previously defined the electric field as the ratio of force/charge, we now define the electric potential as the potential energy/charge: or U = q 0 V U depends on what q o is, but V is independent of q o (can be + or -) Units of electric potential are volts: 1 V = 1 J/C V is a scalar field, defined everywhere in space.

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Work and Potential Energy There is a uniform field between the two plates As the positive charge moves from A to B, work is done W AB =F d=(q E) d ΔU =-W AB =-q E d only for a uniform field

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A proton moves from rest in an electric field of 8.0 10 4 V/m along the +x axis for 50 cm. Find: a) the change in in the electric potential a) the change in in the electric potential b) the change in the electrical potential energy b) the change in the electrical potential energy c) the speed after it has moved 50 cm. c) the speed after it has moved 50 cm. Work and Potential Energy

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b) the change in the electrical potential energy b) the change in the electrical potential energy Work and Potential Energy

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c) the speed after it has moved 50 cm. c) the speed after it has moved 50 cm.

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Potential Difference (=“Voltage Drop”) The potential difference (V) between points A and B is defined as the change in the potential energy (U) (final value minus initial value) of a charge q moved from A to B divided by the size of the charge Potential Difference is not the same as Potential Energy

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To get a positive test charge from lower potential to higher potential you need to invest energy - you need to do work. The overall sign of this: A positive charge would “fall” from a higher potential to a lower one If a positive charge moves from high to low potential, it can do work on you; you do “negative work” on the charge. Electric potential difference, in terms of E Suppose charge q 0 is moved from pt A to pt B through a region of space described by electric field E. A B E F elec F we supply = - F elec Force on the charge due to E work W AB ≡ W A B will have to be done to accomplish this task: q0q0

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Potential Difference and a Battery Electric circuits are all about the movement of charge between varying locations and the corresponding loss and gain of energy which accompanies this movement. The concept of electric potential can be applied to a simple battery-powered electric circuit. Work must be done on a positive test charge to move it through the battery from the negative terminal to the positive terminal. This work would increase the potential energy of the charge and thus increase its electric potential. As the positive test charge moves through the external circuit from the + terminal to the negative terminal, it decreases its electric potential energy and thus is at low potential by the time it returns to the negative terminal. If a 12 volt battery is used in the circuit, then every coulomb of charge is gaining 12 joules of potential energy as it moves through the battery. And similarly, every coulomb of charge loses 12 joules of electric potential energy as it passes through the external circuit.

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Potential Difference and a Battery With a clear understanding of electric potential difference, the role of a battery in a simple circuit can be correctly understood. The battery simply supplies the energy to do work upon the charge to move it from the negative terminal to the positive terminal. By providing energy to the charge, the battery is capable of maintaining an electric potential difference across the two ends of the external circuit. Once the charge has reached the high potential terminal, it will naturally flow through the wires to the low potential

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Potential difference and Energy Question What is the potential difference across an air conditioner if 72C of charge transfer 8.5 x 10 3 J of energy to the fan and compressor?

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Potential difference and Energy Question A static electric shock delivered to a student from a friend transfers 15 J of electric energy through a potential difference of 500V. What is the quantity of charge transferred in the spark?

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Points A, B, and C lie in a uniform electric field. What is the potential difference between points A and B? ΔV AB = V B - V A a) ΔV AB > 0 b) ΔV AB = 0 c) ΔV AB < 0 Point C is at a higher potential than point A? True False E A B C Electric potential

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A positive charge is released from rest in a region of electric field. The charge moves: a) towards a region of smaller electric potential b) along a path of constant electric potential c) towards a region of greater electric potential Electric potential Question

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If you want to move in a region of electric field without changing your electric potential energy. How would you choose the direction ? You would have to move perpendicular to the field if you wish to move without changing electric potential. Electric potential Understanding Question

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Electric Potential Difference A single charge ( Q = -1 C) is fixed at the origin. Define point A at x = + 5m and point B at x = +2m. What is the sign of the potential difference between A and B? ( Δ V AB V B - V A ) x -1 C A B (a) Δ V AB < (b) Δ V AB = (c) Δ V AB > The simplest way to get the sign of the potential difference is to imagine placing a positive charge at point A and determining which way it would move. Remember that a positive charge will always “fall” to lower potential. A positive charge at A would be attracted to the -1 C charge; therefore NEGATIVE work would be done to move the charge from A to B.

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Δ V AB is Independent of Path Δ V AB is the same for any path chosen to move from A to B (because electric forces are conservative). A B q 0 E

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Equipotential lines are connected lines of the same potential. These often appear on field line diagrams. Equipotential lines are always perpendicular to field lines, and therefore perpendicular to the force experienced by a charge in the field. If a charge moves along an equipotential line, no work is done; if a charge moves between equipotential lines, work is done. Electric potential

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Equipotentials and Electric Fields Lines (Positive Charge): The equipotentials for a point charge are a family of spheres centered on the point charge The field lines are perpendicular to the electric potential at all points

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Electric potential Question A direct calculation of the work done to move a positive charge from point A to point B is not easy. Neither the magnitude nor the direction of the field is constant along the straight line from A to B. But, you DO NOT have to do the direct calculation. Remember: potential difference is INDEPENDENT OF THE PATH!! Therefore we can take any path we wish. A positive charge Q is moved from A to B along the path shown by the arrow. What is the sign of the work done to move the charge from A to B? A B (a) W AB < 0 (b) W AB = 0 (c) W AB > 0 Choose a path along the arc of a circle centered at the charge.

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Electric Potential: Where is it zero? Define the electric potential of a point in space as the potential difference between that point and a reference point. a good reference point is infinity... we often set V = 0 Thus, at a distance r from a spherical point charge q, the electric potential is given by:

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Potential from N charges The potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately (this is much easier to calculate than the net electric field, which would be a vector sum). Q1Q1 Q2Q2 Q3Q3 P r 1p r 2p r 3p

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+5 μC -3 μC Two charges q 1 = + 5 μC, q 2 = -3μC are placed at equal distances from the point A. What is the electric potential at point A? a) V A < 0 b) V A = 0 c) V A > 0 A Voltage

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The Electron Volt The electron volt (eV) is defined as the energy that an electron (or proton) gains when accelerated through a potential difference of 1 V Electrons in normal atoms have energies of 10’s of eV Excited electrons have energies of 1000’s of eV High energy gamma rays have energies of millions of eV 1 V=1 J/C 1 eV = 1.6 x 10 -19 J

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Summary Electric potential energy Is the energy of an electrically charged particle In an electric field. Defined as work that must be done to move it from an infinite distance away to its present location. Can be positive or negative (same signs then positive, opposite signs then negative)

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Summary Electric Potential Is the value of the potential energy per unit of positive charge in Volts.

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Summary Electric Potential Difference Is the amount of work required per unit charge to move a positive charge from one point to another in the presence of an electric field. The electrical potential energy for each coulomb of charge in a circuit is called the electric potential difference

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Summary Acts How much work does it take to move a charge of +30 nanocoulombs through a potential difference of 6.0 volts Solution: We are given a potential difference We want work or change in potential energy Therefore:

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Summary Acts Find the speed of the mass of 2.5 x 10 -6 kg if it moves through a potential difference of 30.0 V and has a charge of 4.1 x 10 -6 C. Solution: We require velocity, therefore think about energy (½ mv 2 ). Work =change kinetic energy

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Summary Equations (magnitude only) Electric Field is a Vector quantity (N/C) Electric potential: Where q 0 is the test charge and Q is the charge creating the field Work done Charge density for parallel plates. Potential energy

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Summary Equations n is the number of electrons For parallel plates Electric field of cylinder at a distance r from centre, Is charge density. Capacitance of capacitor Total flux closed circle Any enclosed surface In series, in parallel

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Flash – Electric Potential

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