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PHY2061 Enriched 2 Lecture Notes

Coulomb’s Law

Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to your own notes. Please report any inaccuracies to the professor. Electric

• It is an intrinsic property of particles (i.e. and ) • Comes in both positive and negative amounts (assignment of + and – chosen by Ben Franklin) • Usually denote charge by letter “q”, unit of measure is the Coulomb, C, in SI units −19 o : qee =− =−1.6022×10 C −19 o : qep ==1.6022×10 C o In fact, all charge is quantized in integer multiples of “e” (see further below) • Most is electrically neutral (balanced: equal amounts + and −) o For example, hydrogen, as with all , is neutral. That is lucky for us, otherwise we would have strong attractions to other pieces of matter. But this observation is not explained by any verifiable theory yet! • Can get a net imbalance of electric charge: o on ⇒ excess + charge on glass o Fur on plastic ⇒ excess − charge on plastic • Net charge is always conserved • Like-sign charges repel • Opposite-sign charges attract • Need a law to describe this!

Force Laws

Unit of force is Newtons, N, (kg m s-2), in SI units 2

Newton’s Law of Gravity: r12 mm Fr= G 12ˆ grav 2 12 1 r12 m is mass, measured in kg (SI units). Think of it as the “charge” for gravity. G =×6.67 10−11 N m2 kg-2 is Newton’s gravitational constant r12 is the distance between two masses, i.e. ||r12 rˆ12 is a unit vector pointing along the direction between mass 1 and mass 2

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Note that mass is always defined positive (only one type of gravitational “charge”). Also, the force is always attractive, not repulsive. So the direction of the force is always toward another mass.

Coulomb’s Law (Law of , 1785): qq 12ˆ Frcoul = K 2 12 r12 q is electric charge, measured in C (SI units) 1 K ==91×09 N m2 C-2 is the electrostatic constant [Note: some books use “k”] 4πε 0 −12 2 -1 -2 ε 0 =×8.85 10 C N m is the electric constant

Note that electric charge q can be positive or negative. The force is either attractive (opposite charges) or repulsive (like-sign charges). So the direction of the force is either toward another charge (attractive) or oppositely directed from another charge (repulsive).

That is, the force is always aligned along rˆ12 . Use this guidance in determining the

direction of a force along a particular axis, not the sign of q1× q2 directly.

Interpretation of force: A force causes an object to accelerate if it is free to move.

Newton’s Law: Fa= m

So for the Coulomb force acting on two charged particles otherwise free to move, the acceleration of one of the particles will be:

F K qq 112ˆ ar11== 2 2 mm11r12

Comparison of Gravitational and Electric :

Compare strengths of forces for two objects separated by 1m. Each object has a mass of 1 kg and a charge of 1 C:

11⋅ −11 | FGgrav |==2 6.67×10 N 1 11⋅ | FK|==9×1092 N >10 0×| F| ! coul 12 grav

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Compare attractive force between electron and proton in hydrogen:

9.11××10-31 kg 1.67 10−27 kg mmep −−11 ()( ) 47 | FGgrav |==6.67×10 =4×10 N a 2 −10 2 0 ()0.5×10 m 2 1.6×10−19 C qqep 98()− 40 | FKcoul |==2 ()9×10 N 2 =9×10 N >10 ×| Fgrav | ! a −10 0 ()0.5×10 m

Electric Charge Quantization

Experiment done by American physicist Robert Millikan demonstrated that electric charge is quantized. + + + + + + Millikan’s ⇒ balanced gravitational force, F , with electric force, F Fcoul grav coul q

Fgrav ⇒=qn⋅e n=0,±1,±2,... e=1.6022×10−19 C ------There exists an elementary unit of charge! No smaller charge observed, although “” (constituents of protons and ) are expected to have fractional electric charges. But nevertheless, quantization is still a unique feature. Electrons: q=−e, protons: q= +e. We don’t know why balanced!

1 C 1 Coulomb of electrons is =×61018 electrons! 1.6×10−19 C

Electric

The net sum of electric charge is always conserved. So when a charged conducting object is brought into contact with another conducting object, the charges in the two objects may redistribute, but the net charge of the combined two-object system will remain the same. Likewise, charge is always conserved in reactions:

+− 1 pe+→1 H ee+−+→γγ 238 234 4 92UT→+90 h2 H

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Coulomb’s Law Example (1-dimension)

Consider 2 point charges, q1 and q2, separated by a distance . y Let: qq=−3 q>0 (q is some positive number) F12 F21 1 x qq=− (both charges are negative) 2 q1 q2 x x =1 m

Convention:K F12 denotes the force acting on particle 1 from the presence of particle 2

Since theK force is repulsive (same-sign electric charges), F12 =−F12xˆ (points in negative x direction).

By Newton’s 3rd Law, that for every action (force) there is an equal and opposite reaction,KK the force on particle 2 from particle 1 is: FF21 =− 12 =F12xˆ (points in positive x direction).

Again, use this line of reasoning to determine the direction of the force and not the sign in Coulomb’s Law.

The magnitude of the force is given by Coulomb’s Law: ||qq|| F ==||F KK12=3q2 12 12 x2 Since the force is non-zero and repulsive, the charges will accelerate in the directions specified by the forces.

Can we add a third charge to counteract this force and leave all charge stationary? Yes! Place a third positive charge between charges 1 and 2.

Note that the holds: adding a third charge does not affect the force between charges 1 and 2.

Let particle 1 reside at x = 0, particle 2 at x = 1 m, and particle 3 at x = r, where 0 < r < 1.

y

F12 F13 F23 F21 r q q q x 1 3 2

Find r such that all charges remain at rest, and determine q3 ⇒ 2 unknowns.

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Equilibrium ⇒ F = 0 Sum of all forces acting on particle j equals 0. ∑i ji

For particle 1 this means: FF12 +=13 0 or F12 =−F13 So: qq qq KK12=− 13 r=separation between 1 and 3 122r 2 ⇒=qr32−q

Equilibrium of particle 2 implies: FF21 + 23 ==0 or F21 −F23

qq21 qq23 KK2 =− 2 1−r=separation between 2 and 3 1 ()1− r 2 ⇒=qr31−()1− q

Setting the two equations we obtained for q3 equal yields:

2 2 rq21=−()1 r q Plug in:

qq1 =−3

qq2 =− ⇒=qr 22()12−r +r 3q ⇒−2rr2 6 +3 =0 −±bb2 −4ac quadratic equation ⇒= r 2a 63±−64⋅2⋅331 r ==±3 22⋅ 2 2

1 Only the 0 < r < 1 works, so r =−3 3  0.63 (a little more than half-way) 2 ( ) We can solve for q3: qr=− 2q q=−q 322 ⇒+qq3  0.4 indeed positive

This is an unstable equilibrium. If any charge is moved, the forces will no longer balance and will tend to move the charges even farther away from the equilibrium point (i.e. there is no restoring force to make the equilibrium stable).

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Coulomb’s Law Example (2-dimensions) F23 F21 Consider 3 charges placed in the shape of an equilateral triangle. y q2 Each vertex carries an equal charge: qq= ==q−q (negative) 123 F F and the side length is d. 13 31 q q x 1 3 F F 12 32

Forces on each charge are repulsive (triangle wants to expand). Magnitude of forces are all equal: q2 FF===FF=F=F=K 13 31 12 21 23 32 d 2 But note that directions of the forces are not always the same.

Where can we add a fourth charge q4 so that all charges are at rest? By , we expect the location of a positive fourth charge to be at the center of the triangle. But we need to determine the amount of charge needed to keep the triangle from expanding (forces must balance, so that charges are static).

F23 F21 For particle 1: FF12 ++13 F14 =0 y q2 (condition for forces to balance at equilibrium)

q4 Note that this is a vector equation, and the forces must F13 A F31 separately balance in the x and y directions. q1 q3 x In component notation, these forces are: F12 F32

DD Fx12 =−FF12 cos 60 ˆˆ− 12 sin 60 y

Fx13 =−F13 ˆ DD Fx14 =+FF14 cos30 ˆˆ14 sin 30 y

So balancing each component of the forces gives:

xˆ : −−FFcos60DD+Fcos30 =0 12 13 14 DD yˆ : −+FF12 sin 60 14 sin 30 =0

q2 Now FF==K 12 13 d 2 qq d While F ==K 4 where A 14 A2 3

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Let’s take the equation balancing forces in yˆ :

q2 31qq −⋅KK+ 4 ⋅=0 d 2 22()d 2 /3

−+qq334 =0 q ⇒=qq>0 whereas =−q 413

This solves for the fourth charge, but it is useful to cross-check with the xˆ force equation to check for any mistakes:

qq2213qq −⋅KK− +K4 ⋅=0 dd2222()d 2 /3 33 −+qq30= 224 q ⇒=q 4 3 which checks out. We could also balance the forces acting on any of the other particles as well to determine the fourth charge.

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