Electrical Energy and Potential AP Physics 2

BEWARE!!!!!! W is Electric Potential Energy (Joules) is not V is Electric Potential (Joules/Coulomb) a.k.a Voltage, Potential Difference

Electric Potential Energy Electrical potential energy is the energy contained in a configuration of charges. Like all potential energies, when it goes up the configuration is less stable; when it goes down, the configuration is more stable. The unit is the Joule.

Electric Potential Energy Electrical potential energy increases when charges are brought into less favorable configurations ΔU > 0 + + - +

Electric Potential Energy Electrical potential energy decreases when charges are brought into more favorable configurations. ΔU < 0 + - + +

Electric Potential Energy + – ΔU = ____ ΔU = ____ + + + - Work must be done on the charge to increase the electric potential energy.

Work and Charge For a positive test charge to be moved upward a distance d, the electric force does negative work. The electric potential energy has increased and ΔU is positive (U2 > U1) + d + E F

Work and Charge If a negative charge is moved upward a distance d, the electric force does positive work. The change in the electric potential energy ΔU is negative (U2 < U1) - d F - E

Electric Potential Electric potential is hard to understand, but easy to measure. We commonly call it “voltage”, and its unit is the Volt. 1 V = 1 J/C Electric potential is easily related to both the electric potential energy, and to the electric field.

Electrical Potential and Potential Energy The change in potential energy is directly related to the change in voltage. U = qV U: change in electrical potential energy (J) q: charge moved (C) V: potential difference (V) All charges will spontaneously go to lower potential energies if they are allowed to move.

Electrical Potential and Potential Energy Since all charges try to decrease UE, and DUE = qDV, this means that spontaneous movement of charges result in negative DU. ΔV = ΔU / q Positive charges like to DECREASE their potential (DV < 0) Negative charges like to INCREASE their potential. (DV > 0)

Sample Problem A 3.0 μC charge is moved through a potential difference of 640 V. What is its potential energy change?

Electrical Potential in Uniform Electric Fields The electric potential is related in a simple way to a uniform electric field. V = -Ed V: change in electrical potential (V) E: Constant electric field strength (N/C or V/m) d: distance moved (m) d E DV

Sample Problem An electric field is parallel to the x-axis. What is its magnitude and direction of the electric field if the potential difference between x =1.0 m and x = 2.5 m is found to be +900 V?

Sample Problem What is the voltmeter reading between A and B? Between A and C? Assume that the electric field has a magnitude of 400 N/C. y(m) C 1.0 A B 1.0 2.0 x(m)

Sample Problem How much work would be done BY THE ELECTRIC FIELD in moving a 2 mC charge from A to C? From A to B? from B to C?. How much work would be done by an external force in each case? y(m) C 1.0 A B 1.0 2.0 x(m)

Let’s revisit energy concepts for a gravitational field Let’s revisit energy concepts for a gravitational field. A 250 gram baseball is thrown upward with an initial velocity of 25m/s. What is the maximum height reached by the ball? EARTH v = 25m/s v = 0m/s

Let’s do an example using this new concept: The potential difference between two charge plates is 500V. Find the velocity of a proton if it is accelerated from rest from one plate to the other. High Potential Low Potential 500V + - Positive charges move from high to low potential Negative charges move from low to high potential

A new quantity is defined called the Electric Potential Difference A new quantity is defined called the Electric Potential Difference. It is the electric potential energy difference per unit charge between two points in an electric field. The Electric Potential Difference can also called the Electric Potential, the Potential, or the voltage. Units: Remember Energy and voltage are scalars, so you don’t have to deal with vectors (direction)

Energy per charge The amount of energy per charge has a specific name and it is called, VOLTAGE or ELECTRIC POTENTIAL (difference). Why the “difference”?

Understanding “Difference” Let’s say we have a proton placed between a set of charged plates. If the proton is held fixed at the positive plate, the ELECTRIC FIELD will apply a FORCE on the proton (charge). Since like charges repel, the proton is considered to have a high potential (voltage) similar to being above the ground. It moves towards the negative plate or low potential (voltage). The plates are charged using a battery source where one side is positive and the other is negative. The positive side is at 9V, for example, and the negative side is at 0V. So basically the charge travels through a “change in voltage” much like a falling mass experiences a “change in height. (Note: The electron does the opposite)

Example Calculate the speed of a proton that is accelerated from rest through a potential difference of 120 V 1.52x105 m/s

Let’s make a parallel comparison of Gravitational Potential Energy (Ug) to Electric Potential Energy (UE). Units: Units: UE = Energy that a charge has due to its position in an electric field.

Now we will do an example: An electron is released from rest in an electric field of 2000N/C. How fast will the electron be moving after traveling 30cm? _ v = 0m/s _ v = ? 30cm

Example A pair of oppositely charged, parallel plates are separated by 5.33 mm. A potential difference of 600 V exists between the plates. (a) What is the magnitude of the electric field strength between the plates? (b) What is the magnitude of the force on an electron between the plates? 113,207.55 N/C 1.81x10-14 N

Electric Potential Let’s use our “plate” analogy. Suppose we had a set of parallel plates symbolic of being “above the ground” which has potential difference of 50V and a CONSTANT Electric Field. +++++++++++ DV = ? From 1 to 2 DV = ? From 2 to 3 DV = ? From 3 to 4 DV = ? From 1 to 4 1 25 V 0 V 2 3 d 0.5d, V= 25 V E 12.5 V 4 0.25d, V= 12.5 V 37.5 V ---------------- Notice that the “ELECTRIC POTENTIAL” (Voltage) DOES NOT change from 2 to 3. They are symbolically at the same height and thus at the same voltage. The line they are on is called an EQUIPOTENTIAL LINE. What do you notice about the orientation between the electric field lines and the equipotential lines?

Equipotential Lines So let’s say you had a positive charge. The electric field lines move AWAY from the charge. The equipotential lines are perpendicular to the electric field lines and thus make concentric circles around the charge. As you move AWAY from a positive charge the potential decreases. So V1>V2>V3. Now that we have the direction or visual aspect of the equipotential line understood the question is how can we determine the potential at a certain distance away from the charge? r V(r) = ?

Electric Potential of a Point Charge Why the “sum” sign? Voltage, unlike Electric Field, is NOT a vector! So if you have MORE than one charge you don’t need to use vectors. Simply add up all the voltages that each charge contributes since voltage is a SCALAR. WARNING! You must use the “sign” of the charge in this case.

Potential of a point charge Suppose we had 4 charges each at the corners of a square with sides equal to d. If I wanted to find the potential at the CENTER I would SUM up all of the individual potentials.

Electric field at the center? ( Not so easy) If they had asked us to find the electric field, we first would have to figure out the visual direction, use vectors to break individual electric fields into components and use the Pythagorean Theorem to find the resultant and inverse tangent to find the angle So, yea….Electric Potentials are NICE to deal with! Eresultant

Example An electric dipole consists of two charges q1 = +12nC and q2 = -12nC, placed 10 cm apart as shown in the figure. Compute the potential at points a,b, and c. -899 V

Example cont’ 1926.4 V 0 V Since direction isn’t important, the electric potential at “c” is zero. The electric field however is NOT. The electric field would point to the right.

Applications of Electric Potential Is there any way we can use a set of plates with an electric field? YES! We can make what is called a Parallel Plate Capacitor and Store Charges between the plates! Storing Charges- Capacitors A capacitor consists of 2 conductors of any shape placed near one another without touching. It is common; to fill up the region between these 2 conductors with an insulating material called a dielectric. We charge these plates with opposing charges to set up an electric field.

Electric Potential, Energy, and Capacitance Objective: TSW understand, transfer and apply energy concepts to electric fields and charges by solving problems involving electric fields and forces.

Let’s revisit a gravitational field again: The gravitational potential energy if the field is constant is given by: But, what if the field is not constant. In other words the gravitational potential energy of a mass located far away from the earth.

The gravitational potential energy of mass m is given by: Notice r is not squared The gravitational potential energy at infinity is zero

Let’s make a parallel comparison of Gravitational Potential Energy (Ug) to Electric Potential Energy (UE) when dealing with individual masses or charges. The gravitational potential energy between two masses is given by: The electric potential energy between two masses is given by:

Example: Two 40 gram masses each with a charge of -6µC are 20cm apart. If the two charges are released, how fast will they be moving when they are a very, very long way apart. (infinity)

Let’s use the electric potential energy between two charges to derive an equation for the electric potential (voltage) due to a single point charge. The electric potential energy is given by: q2 P r Let’s remove charge q2 and consider the electric potential (voltage) at point P. q1 To find the potential due to more than one point charge simply add up all the individual potentials: