Electrostatics #4 Energy and Electricity HW #3, last page of handout {2 days set towards this lesson…} HW check (#1 & #2) this Friday!

Note that force and electric field are vector quantities! I. Energy and the Electric Field: Day #1 Introduction: So far, we have concentrated on the interactions of charges through the forces between charges. These interactions can also be modeled through energy, which provides us an explanation of motion from a different point of view. The concepts we will use are: the force acting on a charge by an electric field. the electric field. the charge that is placed into the electric field. The electric field exerts a force on this charge. Note that force and electric field are vector quantities!

Whenever there is a force between two objects, or between an object and a force field, energy can be stored in this interaction. Energy is the ability of an object to perform work. When an object moves under the actions of a force, work is done. For certain forces, this work can be related to stored energy, called potential energy. potential energy. This is the energy stored between two objects due to the force between them. This could also be defined for the force between an object and a field, such as gravitational or electric fields. potential energy due to gravity near the surface of the Earth. A similar expression can be defined for a charged particle interacting with an electric field!

The electric field relates the amount of force available at some location in space, even if there is no charge in that location to feel the force. Similarly Earth’s gravitational field relates the amount of force that could be exerted on some object at a particular location, even if there is no object at that location. The electric field is the force per unit charge available at a given point in space. A similar definition can be made for energy. The electric potential is to potential energy what the electric field is to force. potential energy. This is the energy stored in a charge due to its interaction with an electric field. electric potential. This is the energy per unit charge available in an electric field to interact with a charge.

Units on various quantities… Force: Charge: Electric Field: Potential Energy: Electric Potential: Warning: The variable for Electric Potential is the same as the symbol used for its unit, the volt! For example: be careful of “creative” math! the electric potential is 2 volts, not:

Example #1: Show that the units of electric field can also be written as volts per meter. Note: There is no standard to the units of the electric field, both units may be used. One is not preferred over another.

Derivation of Electric Potential for a Constant Electric Field: It is instructive to see how electric potential is defined and how it relates to the movement of charges in an electric field. Although this is derived for constant electric fields, it generalizes to more complicated arrangements of fields and charges. You will not be required to derive all of this again on homework or on the exam. You are expected to understand the fundamental concepts. If two parallel metal sheets, or plates, are oppositely charged, then (ideally) the electric field between them is constant. The electric field points from the (+) plate to the (–) plate, and it fills the space between the plates. The electric field is confined to the volume between the plates, and is zero everywhere else (ideally).

Start by placing a charge q into an electric field and note the direction of the force on the charge: The force on a positive charge by an electric field points in the same direction as the electric field. The force on a negative charge by an electric field points in the opposite direction as the electric field.

Next consider moving this charge through the electric field Next consider moving this charge through the electric field. The force on the charge by the electric field will do work on the charge q. work, in joules force, in newtons displacement of the charge in the electric field, in meters angle between the force and the motion When the motion is in the same direction as the force, the work done is positive. When the motion is opposite to the force, the work done is negative. Motion ┴ to the force does no work.

Now substitute in specifically the force on a charge in an electric field: work done on the charge q placed in the electric field strength of the electric field, in N/C or V/m amount of charge placed into the electric field. Note: do keep the ± sign on the charge for the calculation. displacement of the charge in the electric field, in meters angle between the electric field and the motion

Since the electric field is a conservative force, a potential energy can be defined for it: difference in potential energy stored in a charge between two different points in an electric field. When a charged particle moves in an electric field, there is a change in the energy stored due to its change of position relative to the field.

Now define the change in electric potential as the change of potential energy per unit of charge: DV represents the change of potential energy per unit charge for a change of position in an electric field. This depends only on the movement relative to the field, an not on the charge interacting with the electric field.

Major Concepts: For the three equations given (work, potential energy, and electric potential), there are three primary cases to know and understand. The three cases are primarily motion at angles of 0º, 90º, and 180º. Case I: 90° An angle of 90º indicates motion perpendicular to the electric field. All three concepts depend on the cosine of 90º, which has a value of zero. When a charged particle moves ┴ to an electric field, no work is done on the charge and the charge does not receive a change to its potential energy. The electric potential also remains constant for any motion ┴ to an electric field.

Case II: 0º This is motion in the same direction as the electric field. The change of electric potential is independent of the charge moving in the electric field, and the value of this is given as follows: Motion in the direction of the electric field lowers the electric potential. The electric field always points from high electric potential to low electric potential. High V Low V direction of electric field

Case II: 0º {continued} If a charged particle moves in the direction of the electric field: When a positive charge moves in the direction of the electric field, the field does positive work on the charge. The charge also lowers its potential energy. The opposite holds true for a negative charge moving in the direction of an electric field: work on the charge by the field is negative and the particle gains potential energy.

Case III: 180º This is motion in the opposite direction as the electric field. The change of electric potential is independent of the charge moving in the electric field, and the value of this is given as follows: Motion opposite to the direction of the electric field raises the electric potential. Again, the electric field always points from high electric potential to low electric potential. High V Low V direction of electric field

Case III: 180º {continued} If a charged particle moves opposite to the direction of the electric field: When a positive charge moves opposite to the direction of the electric field, the field does negative work on the charge. The charge also gains potential energy. The opposite holds true for a negative charge moving opposite to the direction of an electric field: work on the charge by the field is positive and the particle loses potential energy.

Electrostatics #4 Energy and Electricity HW #3, last page of handout Day #2

Review: Constant Electric Field The basic equations used for a particle moving through a constant electric field can be written as: These equations only give the magnitude of the values. Whether the charge has an increase or decrease in any value is determined conceptually. 1. For any motion perpendicular to an electric field , the electric potential V and potential energy PE do not change. No work is done by the electric field. 2. For any motion in the same direction as the electric field, the electric potential decreases. For a positive charge, the potential energy of the charge will decrease. For a negative charge, the potential energy will increase. 3. For any motion opposite to the direction as the electric field, the electric potential increases. For a positive charge, the potential energy of the charge will increase. For a negative charge, the potential energy will decrease.

Ex. #1: A proton moves 2.00 cm parallel to a uniform electric field of E = 200 N/C. (a) How much work is done by the field on the proton? When a positive charge moves in the direction of the electric field, the field does positive work on the charge.

(b) What change occurs in the potential energy of the proton? When a positive charge moves in the direction of the electric field, the potential energy of the particle decreases. (c) What potential difference did the proton move through? Motion in the direction of the electric field decreases the electric potential.

Ex. #2: A potential difference of 90 mV exists between the inner and outer surfaces of a cell membrane. The inner surface is negative relative to the outer surface. How much work is required to eject a positive sodium ion (Na+) from the interior of the cell? inner surface outer surface The force exerted on the ion by the cell is in the direction of motion, so the work done is positive.

Ex. #3: An ion accelerated through a potential difference of 60 Ex. #3: An ion accelerated through a potential difference of 60.0 V has its potential energy decreased by 1.92 × 10−17 J. Calculate the charge on the ion. This is approximately two electron’s worth of charge.

Ex. #4: The potential difference between the accelerating plates of a TV set is about 25 kV. If the distance between the plates is 1.5 cm, find the magnitude of the uniform electric field in the region between the plates. Either unit is acceptable, no one is preferred over the other.

Ex. #5: Oppositely charged parallel plates are separated by 5. 33 mm Ex. #5: 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 between the plates? (b) What is the magnitude of the force on an electron between the plates?

(c) How much work must be done on the electron to move it to the negative plate if it is initially positioned 2.90 mm from the positive plate? The electron is repelled by the negative plate (like charges repel). An outside force must push the electron towards the negative plate. The work done will be positive since the force and motion are in the same direction.

Electric Field from a Point Source Charge: Electric fields are ultimately created by charges. A point charge creates an electric field of: E = magnitude of electric field |Q| = magnitude of source charge r = distance from source charge The corresponding electric potential for this point charge is: Note that the ± sign is kept for the source charge Q, no absolute value. Note also that r is to the first power, not squared.

Electric Field from a Point Source Charge: {continued} The potential energy stored between two charges becomes: Note that the ± sign is kept for both charges, q and Q. Q is considered the source charge, the source of the electric field. q is the charge that interacts with the electric field of the source charge. If the charges are like charges, then the potential energy is positive and the force is repulsive. For opposite charges, the potential energy is negative and the force is attractive.

Ex. #6: (a) Find the electric potential 1.00 cm from a proton.

(b) What is the electric potential difference between two points that are 1.00 cm and 2.00 cm from a proton?

Ex. #7: The three charges in the figure are at the vertices of an isosceles triangle. Let q = 7.00 nC, and calculate the electric potential at the midpoint of the base.

Ex. #8: In Rutherford’s famous scattering experiments that led to the planetary model of the atom, alpha particles (having charges of +2e and masses of 6.64 × 10−27 kg) were fired toward a gold nucleus with charge +79e. An alpha particle, initially very far from the gold nucleus, is fired at 2.00 × 107 m/s directly toward the nucleus, as in Figure P16.19. How close does the alpha particle get to the gold nucleus before turning around? Assume the gold nucleus remains stationary. energy conservation: