Difficulty: Introductory | Prerequisites: Basic algebra, scientific notation, vector addition
This topic sits right at the start of the electricity and magnetism sequence. Everything that follows in PHYS 115, from electric fields to circuits, builds on the idea that matter carries charge and that charges exert forces on each other. You need to be comfortable with charge conservation, the basic mechanics of Coulomb's law, and vector addition of forces before moving on. If you missed the first few lectures, start here.
Electric charge is a fundamental property of matter. Charges come in two types (positive and negative), are conserved in every interaction, and exert forces on each other described by Coulomb's law. The force is proportional to the product of the charges and inversely proportional to the square of the distance between them.
Electric charge
A fundamental property of matter, measured in coulombs (C). Comes in two varieties: positive and negative. Like charges repel, opposite charges attract.
In simple terms, charge is what makes electrical interactions happen. If something has no net charge, it still contains equal amounts of positive and negative charge that cancel out.
Elementary charge (e)
The magnitude of charge carried by a single proton or electron: e = 1.6 x 10^-19 C. All observable charge is an integer multiple of this value.
Think of it as the smallest "unit" of charge that exists in nature.
Coulomb's law
The electrostatic force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them: F = k|q1||q2| / r^2, where k = 8.99 x 10^9 N m^2/C^2.
In simple terms, bigger charges and shorter distances mean stronger forces.
Conductor
A material in which charge moves freely. Metals are the most common conductors. When two identical conducting spheres touch, their total charge redistributes equally between them.
Insulator
A material in which charge does not move freely. Charge placed on an insulator stays where it is put.
Polarisation (charge polarisation)
The rearrangement of charge within an object due to the influence of a nearby charge, without any transfer of charge. A neutral conductor near a charged object will have induced positive and negative regions.
Think of it as charge "shifting" inside an object without leaving it.
Superposition principle (for forces)
The net force on a charge is the vector sum of all individual forces acting on it. Each pairwise Coulomb interaction is calculated independently, then added as vectors.
In simple terms, forces from multiple charges just stack on top of each other.
Charge is always conserved. It cannot be created or destroyed, only transferred.
When two identical conducting spheres are touched together, the total charge distributes equally between them. Each sphere ends up with half the total charge.
To find the charge from a count of excess electrons: multiply the number of electrons by the elementary charge (1.6 x 10^-19 C). Excess electrons means the object is negatively charged.
Example: 2.0 x 10^11 excess electrons = 2.0 x 10^11 x 1.6 x 10^-19 C = 3.2 x 10^-8 C = 32 nC of negative charge. Touch this sphere to a neutral sphere and each ends up with -16 nC.
The magnitude of the electrostatic force between two point charges: F = k|q1||q2| / r^2.
Solving for distance when the force is known: rearrange to r = sqrt(k|q1||q2| / F).
Example: q1 = -3.0 uC, q2 = +5.0 uC, F = 7.5 N. Then r = sqrt(8.99 x 10^9 x 3.0 x 10^-6 x 5.0 x 10^-6 / 7.5) = 0.13 m.
Direction of the force: opposite charges attract (force pulls them together), like charges repel (force pushes them apart). The direction of the force is always along the line connecting the two charges.
When more than two charges are present, calculate the force from each pair separately and then add them as vectors.
To find the net force on a test charge, consider the contribution from every other charge in the system.
The direction of each individual force is determined by whether the interaction is attractive or repulsive.
If the net force on a charge points in a particular direction, work out which arrangement of additional charges could produce that resultant.
A third charge placed at different locations will pull or push the test charge in different directions. Compare the expected force direction at each candidate point with the observed net force to identify the correct location.
Example: two positive charges +q0 are fixed. A negative charge -q0/2 is placed somewhere. The net force on the rightmost +q0 points upward and to the right. The third charge must be above and to the left, at a position where its attractive pull on the test charge, combined with the repulsion from the other +q0, gives the observed direction.
Two objects attract if they carry opposite charges, but also if one is charged and the other is neutral (via polarisation).
Repulsion is the only sure sign that both objects are charged with the same sign.
A neutral conductor near a charged object will have its charge redistributed internally, creating a region of opposite charge closer to the external charge and a region of like charge farther away. The net effect is attraction.
F = k \frac{|q_1||q_2|}{r^2}Coulomb's law. k = 8.99 x 10^9 N m^2/C^2 (Coulomb's constant).
q = n \cdot eCharge from a count of elementary charges. n = number of excess electrons (or protons), e = 1.6 x 10^-19 C.
q_{\text{each}} = \frac{q_{\text{total}}}{2}Charge on each sphere after two identical conductors are touched and separated. The total charge is the sum of the initial charges on both spheres.
Coulomb's law is the basis for understanding how electrostatic precipitators remove soot and particulates from industrial exhaust: charged particles are attracted to oppositely charged plates and pulled out of the air stream. The same charge-sharing principle behind touching conducting spheres is at work every time you get a static shock from a doorknob after walking across carpet.
Students often think that when two spheres share charge, the neutral sphere gets all the charge. It does not. The total charge splits equally between two identical conductors.
Students often assume attraction means both objects must be charged. It does not. A charged object will attract a neutral conductor through polarisation.
Students sometimes forget that Coulomb's law gives only the magnitude of the force. The direction must be determined separately from the signs of the charges (like repels, opposite attracts).
Students frequently mix up the sign of the charge when converting from electron count. Excess electrons means negative charge, not positive.
Charge-sharing between identical conductors is a staple exam question. Know the formula and watch the sign.
Expect to be asked to solve Coulomb's law for distance, given force and charges, or vice versa.
Superposition problems with three or more charges appear frequently. You need to be comfortable adding force vectors.
Questions that ask "is the object charged, and if so, what sign?" are testing whether you understand that attraction alone is not proof of charge.
True or False: When two identical conducting spheres with charges +10 nC and -6 nC are touched and separated, each sphere ends up with +2 nC. (True)
True or False: Coulomb's law applies only to positive charges. (False: it applies to any pair of point charges.)
Fill in the blank: The force between two charges doubles if the distance between them is reduced by a factor of ___. (sqrt(2), i.e. about 1.41)
True or False: If object A attracts object B, both must be charged. (False: B could be neutral and polarised.)
Fill in the blank: One electron carries a charge of ___ C. (1.6 x 10^-19)
Q: Sphere A has an excess of 2.0 x 10^11 electrons and Sphere B is neutral. Both are identical metal spheres. They are touched together and separated. What is the charge on Sphere B afterwards?
A: The total charge is -(2.0 x 10^11)(1.6 x 10^-19) = -32 nC. After contact, each sphere gets half: -16 nC.
Q: Two charges q1 = -3.0 uC and q2 = +5.0 uC are separated by a distance d. The net force on q1 is 7.5 N, directed toward q2. What is d?
A: Use F = k|q1||q2|/d^2. Rearrange: d = sqrt(k|q1||q2|/F) = sqrt(8.99 x 10^9 x 3.0 x 10^-6 x 5.0 x 10^-6 / 7.5) = 0.13 m.
Q: Two metal balls attract each other. Ball A is fixed on an insulating stand. Ball B hangs from an insulating string. Is Ball A necessarily charged? If so, what sign?
A: There is not enough information to determine the sign (or even whether Ball A is charged). Attraction can occur between two oppositely charged objects, or between a charged object and a neutral object via polarisation. Ball B could be charged and Ball A neutral, or they could carry opposite charges.
Q: A point charge Q0 is placed above a partial circular insulating rod carrying charge +QR. A positive test charge +q0 at the centre of the rod feels zero net force. What is the sign of Q0, and how does |Q0| compare to |QR|?
A: The missing section of the rod means the rod's field at the centre points away from the gap (toward the existing arc). Q0 must be negative (to attract +q0 back toward the gap side and cancel the rod's force). Because Q0 is a point charge at some distance while the rod is distributed at the same radius, and because part of the rod is missing, |Q0| is less than |QR|. The rod contributes a net force only from the asymmetry of the missing piece, so Q0 need only cancel that smaller residual force.
Coulomb's law is the foundation for the electric field concept covered next: the electric field at a point is just the Coulomb force per unit test charge. Superposition of forces leads directly to superposition of electric fields. Understanding charge conservation here also matters for circuit analysis later, where charge flowing per second defines current.
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