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Four charges in a squareFour charges of equal magnitude are placed at the corners ofa square that measures L on each side. There are twopositive charges Q diagonally across from one another, andtwo negative charges -Q at the other two corners.

Four charges in a squareFour charges of equal magnitude are placed at thecorners of a square that measures L on each side.There are two positive charges Q diagonally acrossfrom one another, and two negative charges -Q atthe other two corners.How much potential energy is associated with thisconfiguration of charges?1. Zero2. Some positive value3. Some negative value

Four charges in a square1. Determine how many ways you can pair up the charges.For each pair, write down the electric potential energyassociated with the interaction.kQ 2We have four terms that look like: LkQ 2And two terms that look like: 2LAdd up all the terms to find the total potential energy. Do weget an overall positive, negative, or zero value?

Four charges in a square1. Determine how many ways you can pair up the charges.For each pair, write down the electric potential energyassociated with the interaction.kQ 2We have four terms that look like: LkQ 2And two terms that look like: 2LAdd up all the terms to find the total potential energy. Do weget an overall positive, negative, or zero value? Negative

Four charges in a square2. The total potential energy is the work we do to assemble theconfiguration of charges. So, let’s bring them in (from infinity)one at a time.It takes no work to bring in the charge 1.Bringing in - charge 2 takes negative work, because we haveto hold it back since it's attracted to charge 1.

Four charges in a square2. The total potential energy is the work we do to assemble theconfiguration of charges.Bringing in the charge 3 takes very little work, since there'salready one charge and one – charge. The work done is alsonegative because it ends up closer to the negative charge.Bringing in the - fourth charge also takes negative workbecause there are two positive charges and one negativecharge, so overall it's attracted to them.The total work done by us is negative, so the system hasnegative potential energy.

A charge and a dipoleA dipole is placed on the x-axis with its center on the origin. Apositive point charge will be moved from very far away on they-axis to the origin. In Case 1, it will be moved straight downthe y-axis. In Case 2, it will follow a complicated path but itsstarting and ending points will be the same as in case 1.Which case takes more work?Case 1Case 2

A charge and a dipoleWhich case takes more work?1. Case 12. Case 23. The work done is the same in both casesCase 1Case 2

A charge and a dipoleLike gravity, the electrostatic force is conservative. When theonly forces acting are conservative, it doesn't matter how anobject gets from A to B, the work done is always the same.

How much work?How much work is required to bring the charge from veryfar away to the center of the dipole?1. Zero2. The work done is positive3. The work done is negative

How much work?The potential at the two end-points is the same, zero. Thechange in potential energy is: U q V 0The work done by the field is - U. We would have to do anamount of work U to bring in the charge against the field,but, because U 0, no work is done.

The point is special, not the chargeOur conclusion, that no net work is done to move a charge(any charge) from far away to the place halfway between thetwo charges in the dipole, shows us that the point we aremoving the charge to is special.Something about the combined influence of the two chargeson that point is zero. What is zero for that point?

Electric potentialToday, we focus on electric potential, which is related topotential energy in the same way electric field is related tovforce.v FU UV V E qqqElectric potential, like field, is a way to visualize how acharged object, or a set of charged objects, affects the regionaround it.A voltage is essentially a difference in electric potential, whichchanges how charges flow in a way analogous to howpressure differences affect the flow of fluid.

Visualizing electric potentialWe often draw equipotentials (lines of constant potential) on apicture involving charges and/or fields. An equipotential isanalogous to contour lines on a map, such as this map of thesummit of Mt. Rainier. What do the contour lines represent?Lines of constant.Photo credit:NASA/USGSField lines are always perpendicular to equipotential lines.

Visualizing electric potentialWe often draw equipotentials (lines of constant potential) on apicture involving charges and/or fields. An equipotential isanalogous to contour lines on a map, such as this map of thesummit of Mt. Rainier. What do the contour lines represent?Lines of constantheight.Photo credit:NASA/USGSField lines are always perpendicular to equipotential lines.

Equipotentials in a uniform fieldHere’s a picture of equipotentials in a uniformelectric field.In which directionis the electric field?

Equipotentials in a uniform fieldHere’s a picture of equipotentials in a uniformelectric field.In which directionis the electric field?Down – field pointsin the direction ofdecreasing potential.Also, the units of J/Care equivalent to the volt (V).v VE r

Electric potential in a uniform fieldPotential difference, V, is far more important than potential.In a uniform electric field: U F r cosθ qE r cosθ V E r cosθqqqwhere θ is the angle between the field and the displacement.When we just need the magnitude of the potential difference,we often simplify the above to V Ed ,where d is the distance moved parallel to the field.The analogous gravitational situation is:Gravitational potential difference U mgh Vg ghmm

Moving through the fieldA q test charge is moved vertically a distance r in theregion of uniform field. What is the change inpotential experienced by this charge?1. Zero2. kq / r3. –kq / r4. 12 volts5. -12 voltsEndStart

Moving through the fieldThe charge moves from the -4 V line to the 8 V line, for anet change in potential of 12 V.EndStart

A negative charge?How would your answer change if the charge had been anegative charge, -q, instead?1. The answer would not change2. The answer would flip sign

A negative charge?It doesn’t matter what moves from the -4 V line to the 8 Vline, the net change in potential is still 12 V.When you flip the sign of the charge, what does reversesign is the change in potential energy. U q V

Potential from a point chargeThe electric potential set up by a point charge is an exampleof potential when the field is non-uniform. Note that thepotential is defined to be zero whenr infinity.Electric potential a distance r from a point charge : V In which direction is the electric field in the picture?kqr

Which way is the field?The simulation shows the equipotentials for a non-uniformfield, specifically the field from a point charge. Inwhich direction is the field?1. Clockwise2. Counter-clockwise3. Toward the center4. Away from the center5. There is not enough information to say

Which way is the field?Field points in the direction of decreasing potential. In thiscase, that is toward the charge.You can also recognize that this pattern of equipotentials isproduced by an object with a negative charge, and theelectric field points toward a negative charge.

Worksheet: where is the potential zero?Two charges, 3Q and –Q, are separated by 4 cm. Is there apoint along the line passing through them (and a finitedistance from the charges) where the net electric potential iszero? If so, where?First, think qualitatively.Is there such a point to the left of the 3Q charge?Between the two charges?To the right of the –Q charge?

Worksheet: where is the potential zero?Unlike electric field, where we had to worry about two vectorsbeing equal and opposite, we just have to worry about twonumbers having the same magnitude but opposite sign andthey automatically have opposite signs.One charge has three times the magnitude of the other. Thus,we’re looking for points that are times farther from the 3Q charge than the –Q charge.In which region(s) can we find such points?

Worksheet: where is the potential zero?Unlike electric field, where we had to worry about two vectorsbeing equal and opposite, we just have to worry about twonumbers having the same magnitude but opposite sign andthey automatically have opposite signs.One charge has three times the magnitude of the other. Thus,we’re looking for points that are three times farther from the 3Q charge than the –Q charge. (V kq / r)3kQ kQ 0r1r2In which region(s) can we find such points?Region II and Region III.

Worksheet: where is the potential zero?The two charges are separated by 4 cm.At what location between the charges is the net electricpotential equal to zero?At what location to the right of the –Q charge is the netelectric potential equal to zero?

Worksheet: where is the potential zero?The two charges are separated by 4 cm.At what location between the charges is the net electricpotential equal to zero?1 cm from the –Q charge, and 3 cm from the 3Q charge.At what location to the right of the –Q charge is the netelectric potential equal to zero?2 cm from the –Q charge, and 6 cm from the 3Q charge.

Off the line?On the straight line passing through the charges there isonly one location (a finite distance from the charges)where the net electric field is zero. There are two placeson the line where the net potential is zero. Are there areplaces that are not on the straight line joining the charges,a finite distance away, where the field and/or the potentialis zero?1. No2. Yes for both.3. Yes for field, No for potential.4. No for field, Yes for potential.

Worksheet: where is the potential zero?

Making up questionsTwo charges, 3Q and –Q, are separated by 4 cm. Thecharges are on the x-axis, with the 3Q charge at x -2 cmand the –Q charge at x 2 cm.Ask a question involving force for this situation.How much force does the 3Q charge feel? (There are veryfew questions like this.)Ask a question involving field for this situation.What is the net electric field at the point x 3 cm, y 5cm? (There are an infinite number of questions like this!)Ask a question involving field, and then a follow-up questioninvolving force.What is the net field at the origin? How much force does a 2Q charge experience when placed at the origin?

Making up questions, IITwo charges, 3Q and –Q, are separated by 4 cm. Thecharges are on the x-axis, with the 3Q charge at x -2 cmand the –Q charge at x 2 cm.Ask a question involving potential energy for this situation.Then re-phrase it without the words “potential” or “energy.”Ask a question involving potential for this situation.Ask a question involving potential, and then a follow-upquestion involving potential energy.

Making up questions, IITwo charges, 3Q and –Q, are separated by 4 cm. Thecharges are on the x-axis, with the 3Q charge at x -2 cmand the –Q charge at x 2 cm.Ask a question involving potential energy for this situation.Then re-phrase it without the words “potential” or “energy.”What is the potential energy of this pair of charges?How much work was done to assemble this set of charges?Ask a question involving potential for this situation.What is the electric potential on the x-axis at x 5 cm?(There are an infinite number of questions like that.)Ask a question involving potential, and then a follow-upquestion involving potential energy.What is the electric potential on the y-axis at y 2 cm? Whatis the potential energy of a –2Q charge placed there?

Electric field near conductors, at equilibriumA conductor is in electrostatic equilibrium when there is nonet flow of charge. Equilibrium is reached in a very short timeafter being exposed to an external field. At equilibrium, thecharge and electric field follow these guidelines: the electric field is zero within the solid part of the conductor the electric field at the surface of the conductor isperpendicular to the surface any excess charge lies only at the surface of the conductor charge accumulates, and the field is strongest, on pointyparts of the conductor

Electric field near conductors, at equilibriumAt equilibrium, the field is zero inside a conductor andperpendicular to the surface of the conductor because theelectrons in the conductor move around until this happens.Excess charge, if the conductor has a net charge, can onlybe found at the surface. If any was in the bulk, there wouldbe a net field inside the conductor, making electrons move.Usually, the excess charge is on the outer surface.

Electric field near conductors, at equilibriumCharge piles up (and the field is strongest) at pointy ends ofa conductor to balance forces on the charges. On a sphere,a uniform charge distribution at the surface balances theforces, as in (a) below.For charges in a line, a uniform distribution (b) does notcorrespond to equilibrium. Start out with the charges equallyspaced, and the forces the charges experience push themso that they accumulate at the ends (c).

A lightning rodA van de Graaff generator acts like a thundercloud. We willplace a large metal sphere near the van de Graaff and see whatkind of sparks (lightning) we get. We will then replace the largemetal sphere by a pointy piece of metal. In which case do weget more impressive sparks (lightning bolts)?1. with the large sphere2. with the pointy object3. neither, the sparks are the same in the two cases

A lightning rodThe big sparks we get with the sphere are dangerous, and inreal life could set your house on fire.With the lightning rod, the charge (and field) builds up soquickly that it drains charge out of the cloud slowly andcontinuously, avoiding the dangerous sparks.The lightning rod was