Electric Potential Unit 2. Energy in Physics Energy can be used to study a wider range of phenomena...

Electric PotentialUnit 2

Energy in Physics

• Energy can be used to study a wider range of phenomena and problems that cannot be easily analyzed using forces.

• Energy is also nice because it is a scalar rather than a vector.

• In this unit, we will look at how energy is applied in electrostatics.

Electric Potential Energy

• Recall from last year that the change in an object’s potential energy between two points is equal to the negative work done by the conservative force acting on the object.

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ΔU =Ub −Ua = −Wba

Example: Proton in an Electric Field

A proton is placed in a uniform electric field of 20000 N/C. It experiences a force to the right.

a) What is the magnitude of the force being exerted on the proton by the field?

b) How much work does the field do on the proton if it moves the proton 3.0 cm?

c) What was the change in the proton’s potential energy?

This is actually a general result for a particle in a uniform electric field.

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ΔU = −qEd

Electric Potential Energy

• Notice that, just like with gravitational potential energy, we are free to set U = 0 wherever we please.

• Therefore, it is only meaningful to talk about differences in potential energy.

• This we will also be true for electric potential.

Electric Potential

Electric Potential

• In the last unit, we ran into a problem with the “action-at-a-distance” idea of a force.

• This problem would translate to energy, since it is related to force.

• To deal with this, we introduced the electric field, which is the force per unit charge.

Electric Potential

• Similarly, we will introduce electric potential to help deal with energy.

• Electric potential is the electric potential energy per unit charge.

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Va =Ua

qElectric potential at point a

Electric PE at point a

Charge of the object

Electric Potential• Like potential energy, electric potential

must be defined relative to a zero.

• That means that only differences in potential have any real meaning.

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Vba =Vb −Va =Ub −Ua

q

Electric Potential

• We prefer to use electric potential instead of potential energy because V (like E) does not depend on the size of the test charge.

• V is created by the charges creating E, and q gains potential energy by being placed in the potential.

Properties of Potential

• Positive charges move from high V to low V.

• Negative charges do the reverse.• Potential is has units of volts (V).

• The word voltage is used to mean potential difference.

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1V =1JC

Problem Solving

• Just as we often found the electric force from the electric field, we will generally use voltage to find electric potential energy.

• This can be done by rearranging our definition of potential.

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ΔU = q Vb −Va( ) = qVba

Zero Volts• We can define V = 0 at any place.

However, there are conventions for where to chose.

• One common choice is to set 0V at the ground. Any object that is grounded is also at 0V.

• The other choice is to set V = 0 at infinity. More on this next week.

The Electron Volt

• We noticed yesterday that the amount of energy involved in the movement of electrons and protons is very small.

• The joule is too large a unit to effectively describe the energy of electrons.

• Notice that we get units of energy when we multiply a charge times a potential difference.

The Electron Volt

• We can use this fact to define a new unit of energy.

• An electron volt (eV) is the amount of energy an electron receives when moving through a potential of 1 volt.

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1eV =1.6 ×10−19J

Example: Moving a Proton• How much work is done by the electric

field moving a proton from a 100 V to -20 V?

Electron in a TV TubeThe picture tube of a television set has a potential difference of 5000 V. An electron is accelerated from rest through this potential difference.

a) What is the change in potential energy of theelectron?

b) What is the speed of the electron when it reachesthe end of the tube?

- +

5000 V

Homework

• Do problems 1-4, 8, and 11 on page 489.

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Potential vs. Potential Energy

An Analogy• Suppose we decided to drop a rock off

a cliff.

• The higher the cliff, the more potential energy the rock would have (U = mgh).

• Remember, potential energy is a measure of how much KE the object would gain as a result of the work done by gravity.

An Analogy• The actual amount of KE the object

gains depends not only on height, but also on the mass of the object.

• We could drop a large rock and a small rock from the same height, but the large rock would have more PE.

• The height would be considered to be the gravitational potential.

An Analogy• The electrical case is similar. Two

charged particles can be at the same location in an electric field. They are at the same potential.

• However, the particle with the greater charge has more potential energy.

• One key difference between charge and mass: charge can also be negative.

Potential and the Electric Field

Potential and the E Field

• Although we did not define V this way, it is very closely related to E.

• Mathematically, this relationship is

• Generally, this expression is difficult to evaluate without the aid of calculus.

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limΔx→0

E = −ΔVΔx

Potential and the E Field

• However, we can see how V and E are related for a constant electric field.

• Recall the following equations:

• So, the work done by the electric force is:

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ΔU = qVbaW = −ΔU

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W = −qVba

Potential and the E Field• However, since the E field is constant

between the plates, we also know:

• Where here

• So€

W = Fdcosθ

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F = qE θ = 0o

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W = qEd

Potential and the E Field• So, now we have two expressions for

work:

• Setting these equal, we get:

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W = −qVba

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W = qEd

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Vba = −Ed

E = −Vbad

ONLY true if the E field is uniform

Example: Find the E Field

- +

50 V

Two parallel plates are separated by a distance of 0.05 m and are charged to produce a potential difference of 50 V between them.

What is the magnitude of the electric field between the plates?

0.05 m

Problems

• Do problems 5-8, 10, and 11 on page 489 of the book.

• Your homework is to finish these problems.

A quick word about E

• Yesterday, we saw that a constant electric field is related to potential by

• In this equation, E has units of V/m. This is an alternative way to express E, and it is equivalent to the N/C formulation.€

E = −Vbad

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1NC=1Vm

Potential of a Point Charge

Potential due to a Point Charge

• The potential a distance r away from a point charge Q can be found from the charge’s electric field using calculus.

• The result is

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V =kQr

Here, V = 0 at r = .

Potential due to a Point Charge• Notice that while E falls off

as 1/r2, V falls off as 1/r.

• The sign of the charge is important.– V is very large and positive

near a positive charge, and decreases to zero far away.

– V is very large and negative near a negative charge, and increases to zero far away.

QuickTime™ and a decompressor

are needed to see this picture.



Example: Potential of a Point Charge

What is the potential 0.5 meters away from a point charge magnitude +20 C?

Repeat the problem but make the charge -20 C.

Example: Bringing Two + Charges Together

How much work is needed to bring a 3 C test charge from a great distance away (say ) to a point 0.5 m away from a +20 C charge?

Adding Potentials

• In the last unit, we learned that, to find the net electric field, you add up the individual fields caused by source charges.

• Since E is a vector, you must use vector addition.

• This is annoying.

Adding Potentials

• Luckily, potential is a scalar.

• This means you can add potentials just like regular numbers, without worrying about direction.

• You must include the sign of the source charges however.

Example: Adding PotentialsFind the electric potential at points A and B due to the two source charges shown.



Problems

• Do problems 14, 15, 19, and 23 on page 489.


Problem Day

• Do problems 9, 13, and 16 on page 489.

• We will whiteboard these in 30 mins.

• If you finish early, work on problem 65 on page 491.

Whiteboarding Groups

Group Members Problem1 John, Brie 142 Sarah, Drew, Angi 153 Kaleb, Jeremiah, Bailey 194 Miggy, Connor, Armen 235 Anthony, Abbey, Aidan 96 Rachel, Jacob, Piper 137 Robert, Ellen, Krystiana 16

Homework

• Read section 17-3

• Do problem 65 on page 491.

Equipotential Lines

Equipotential Lines• Earlier in the unit, we likened potential

to “height” of the electric field.

• When we diagram heights in space, we sometimes use a contour map.

• A contour line indicates an elevation above sea level.

• Any point on that line is at that height.

Equipotential Lines



Equipotential Lines

• We can do the same thing for the electric field.

• Contour lines for the E field are called equipotential lines.

• All points on an equipotential line are at the same potential.

Equipotential Lines

• Since the work required to move a charge between two points depends on the voltage difference, no work is needed to move a charge along an equipotential line.

• No work is also needed to move a charge perpendicular to the electric field.

Equipotential Lines

• Therefore, equipotential lines must be perpendicular to the E field at all points.

• The E field always points towards lower values of V.

• The closer together the lines are, the stronger the electric field is in that area.

Equipotential Lines



Equipotential Lines



Potentials and Conductors

• Since the E field inside a conductor is zero, the conductor must be entirely at the same potential.

• A conductor is an equipotential.

• Incidentally, this is why E field lines are always perpendicular to the surface of a conductor.

Homework

• Read section 17-7

• Do problems 17 and 65 on page 489.

Capacitance and Dielectrics

Capacitors

• A capacitor is a device used to store electrical energy.

• They come in a variety of shapes and sizes.

• Usually, a capacitor consists of two sheets of metal separated by a small distance.

Capacitors

• Capacitors have a wide range of uses, from blocking electric surges, to camera flashes, to computer RAM.





Capacitors

• If a voltage difference is applied across a capacitor (one plate is forced to be at a higher potential than the other), the two plates of the capacitor become charged.

• One plate has a negative charge, the other an equal amount of positive charge.

Capacitors

• The amount of charge is proportional to the voltage applied. In other words:

• The constant, C is called the capacitance of the capacitor.

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Q =CV

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C =QV

Capacitance

Charge on each plate

Applied voltage

Capacitance

• The unit of capacitance is a coulombs per volt.

• This unit is called a farad (F).

• Typical capacitors have capacitances in the range of 1 pF (picofarads) to ~100 F (microfarads).

Parallel Plate Capacitors

• Capacitance, in general, does not depend on Q or V.

• How much charge can fit on the capacitor depends on the size of the plates.

• The voltage difference depends on the separation of the plates.

Parallel-Plate Capacitors

• We can use these observations to write down the capacitance for a parallel-plate capacitor with plates of area A, separated by a distance d.

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C = ε 0Ad

Parallel-plate capacitor

only.

Area of each plateDistance between plates

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ε0 = 8.85 ×10−12 C 2

N ⋅m 2

Example: Capacitance

A parallel-plate capacitor is made by placing two 20 cm x 3 cm plates 1 mm apart.

a) What is the capacitance?

b) If the capacitor is connected to a 12-V battery, what is the charge on each plate?

c) What is the magnitude of the E field between the plates?

d) What would the area of the plates need to be to give a capacitance of 1F?

Homework

• Do problems 33-39 odd on page 490.

• Check your answers in the back of the book.

Circuit Symbols

• The capacitor is the first of many circuit elements we will study this year.

• As it can be tedious to draw pictures of actual circuits, we will be using circuit diagrams to represent how circuits are set up.

• Let’s go over some circuit symbols.

Circuit Symbols

• The simplest capacitor is just two parallel plates separated by a small distance.

• Capacitors are represented by the symbol:

Circuit Symbols

• A battery is a device that maintains a potential difference between two points (terminals).

• We will discuss this more next unit.• Batteries are represented with the

symbol:+

Circuit Symbols

• A wire is a conductor used to connect two circuit elements.

• Any two points on a wire are at the same voltage.

• Wires are represented by lines in circuit diagrams.

Circuit Symbols



Dielectrics

Dielectrics

• The space inside most capacitors are is not filled with air.

• Instead, the space is filled with an insulating material.

• These materials are known as dielectrics.

Dielectrics

• There are two main reasons for using a dielectric in a capacitor.

1. The plates can be put closer together, because the dielectric keeps the charge from jumping across the gap.

2. The dielectric itself increases the capacitance.

Dielectrics







Dielectrics• When you place a dielectric in a

capacitor, the capacitance is increased by a factor of K.

• K is called the dielectric constant of the material.

• The value of K can be found in a table.€

C =Kε 0Ad

Dielectrics

• The formula for capacitance can also be written:

• Where

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C = εAd

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ε =Kε 0Permittivity of the material.

Application: Computer Keyboards



Homework

• Read sections 17-8 and 17-9.

• Do problems 42-45 on page 490.

• We will be having our next test this Friday.


Group Members Problem1 John, Brie 422 Sarah, Angi 433 Kaleb, Jeremiah, Bailey 444 Miggy, Jacob 455 Anthony, Abbey, Aidan 336 Rachel, Piper, Drew 357 Connor, Armen 378 Robert, Ellen, Krystiana 39

Dielectric Breakdown




• We learned earlier this year that dielectrics (insulators) prevent charge from flowing.

• This is only true below if the voltage difference across the material is below a certain level.

• If the voltage is too high, the material becomes ionized and charge is able to flow.


• To illustrate this, we will use the phenomenon of lightning.

• During a storm, clouds acquire a net negative charge.

• The process by which this happens is not agreed upon by all scientists.


• The Earth is a conductor and the negative charge of the cloud repels electrons from the surface of the Earth.

• The surface has a net positive charge.

• This leads to an electric field similar to that of a parallel-plate capacitor.


• The voltage difference between the cloud and the Earth creates a force attracting the electrons towards the Earth.

• The electrons are prevented from flowing by the air, which is an insulator.


• However, if the voltage is large (~10,000 volts/in2 for air), the force is strong enough to pull electrons away from the air molecules.

• The air becomes ionized.

• Ions are excellent conductors.


• The electrons in the cloud now have a clear conductive path and discharge into the ground.

• This reduces the potential difference between cloud and ground.

• The voltage is no longer strong enough to ionize the air, and no more charge is able to flow until the charge builds up again in the cloud.


• Dielectric breakdown can occur in any insulating material.

• The voltage needed for breakdown to occur depends on the material.

• For this reason, capacitors are rated for a maximum voltage.



Storage of Electrical Energy


• We defined a capacitor as a device for storing electrical energy.

• We now want to describe how much energy is stored in the capacitor.

• If a capacitor is discharged, the charge flows off the plates and can do work (in an electric circuit).

• Therefore, this is potential energy.


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U = 12QV

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U = 12CV

2

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U = 12Q2

C

Q - Charge on each plate of the capacitor.

V - Potential difference across the capacitor.

C - Capacitance of the capacitor.


• It is interesting to note that the energy in these equations is stored in the electric field between the plates of the capacitor.

• This suggests that any electric field contains potential energy.


• For a regular shape (like a parallel-plate) capacitor, it is fairly easy to quantify the amount energy stored in the field.

• This is only true for a parallel-plate capacitor.

• See section 17-9 for the derivation.

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U = 12 ε 0E

2Ad


• However, it is usually more useful to talk about energy density.

• This is the amount of energy stored in the field per unit volume.

• This is true for any electric field.

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energy density = 12 ε 0E

2

Homework

• Do problems 46-49 on page 490.

• You may want to refer to section 17-9 while doing these problems.

Announcements

• We will be having our next test this Friday.

• I will give you a formula sheet on the test.

• The AP review sessions begin this Friday after class.

• I will start Tuesday morning tutoring next week.

Problem Day

• Do problems 60, 61, 67, 70, 73, and 75 on page 491-492.



Group Members Problem1 John, Angi, Ellen,

Krystiana60

2 Brie, Armen, Aidan 613 Kaleb, Jeremiah, Bailey 674 Miggy, Sarah, Abbey 705 Anthony, Drew, Connor 736 Rachel, Jacob, Piper,

Robert75

The Electrocardiogram

The Heart

• The human heart is responsible to pumping blood to the rest of the body.

• It is electrical impulses that make the heart beat.

• These electrical impulses can be detected by an electrocardiogram (EKG).

The Heart



The Heart

• When the heart is at rest (in between beats), there is a natural separation of charge between the inside of heart cells and the outside.

• This separation of charge leads to a potential difference across the cell membrane.



The Heart

• The inside of a heart cell is at a potential that is about 90 mV less than the outside of the cell.

• When the muscle contracts, the membrane allows the positive ions to pass inside the cell.





The Heart• This “depolarization” of

the cell starts at one side and progresses across the cell.

• The charge distribution of the cell has now changed.

• This means the voltage difference has also changed.





The Heart• Once the cell is

depolarized, the process reverses and the cell re-polarizes to its original state.

• The changes in voltage are read by the EKG and analyzed to determine the health of the heart.





The EKG

• The EKG graphs the voltage between two points on opposite sides of the heart as a function of time.

• A typical EKG graph is consists of three “waves” that represent a single heartbeat.



The EKG



P wave

Contraction of the atria

QRS complex

Contraction of the ventricles

T wave

Re-polarization of the heart

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Electric Potential Unit 2. Energy in Physics Energy can be used to study a wider range of phenomena...

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