Coulomb's Law

Coulomb's Law

• Describes the electrostatic interaction between electrically charged particles.
• Studied and published in 1783 by French physicist Charles Augustin de Coulomb.
• Essential to the development of electromagnetism theory.
• Scalar form describes the magnitude of electrostatic force between two electric charges.
• Vector form is needed to describe the direction of the force.
• The magnitude of electrostatic force (F) on a charge (q1) due to the presence of a second charge (q2) is: F=k₀ * (|q1q2|/r²).
• r is the distance between the two charges
• k₀ is a proportionality constant (Coulomb's constant).
• k₀ = 1/(4πε₀) = c²µ₀ = c² * 10⁻⁷ H/m
• k₀ ≈ 9 x 10⁹ N⋅m²/C²
• The speed of light in a vacuum (c) is approximately 3 x 10⁸ m/s.
• The magnetic constant (µ₀) equals 4π x 10⁻⁷ H/m.
• The electric constant (ε₀) is approximately 8.85 x 10⁻¹² F/m.
• The magnitude of the force is directly proportional to the magnitudes of the charges and inversely proportional to the square of the distance between them.

Electric Field

• An electric field is a region of space where a charge experiences an electric force.
• Produced by one or more charges, and can be uniform or vary in magnitude and direction.
• If a charge q at a point experiences a force F, the electric field E at that point is defined as: E = F/q.
• According to the Lorentz Force Law, the magnitude of the electric field (|E|) created by a single point charge (q) at a distance (r) is: E = 1/(4πε₀) * q/r².
• The electric field points along lines directed radially away from the location of the point charge for a positive charge.
• The electric field points in the opposite direction for a negative charge.
• The SI units of electric field are volts per meter (V/m) or Newtons per coulomb (N/C).

Electric Field Lines

• Electric field lines are imaginary lines that indicate the direction and magnitude of the electric field.
• The direction of an electric field line at any point shows the direction a positive charge would move if placed there.
• Field lines are closer together where the field is strong and farther apart where the field is weak.

Electric Potential Difference

• The potential difference between points A and B is the work done against electrical forces in carrying a unit positive test charge from A to B.
• Potential difference between A and B is represented as Vb - Va, or just V (when there is no ambiguity).
• Units are those of work per charge, designated as volts.
• 1 V = 1 J/C
• Work (W) done in transporting a charge q from point A to point B: W = q(Vb - Va) = qV
• Appropriate sign must be given to the charge.
• If both Vb - Va and q are positive or negative, the work done is positive.
• If Vb - Va and q have opposite signs, the work done is negative.

Absolute Potential

• The absolute potential at a point is the work done against electric forces in carrying a unit positive test-charge from infinity to the point.
• Absolute potential at point B is the difference in potential from A at ∞ to B.
• Absolute potential at point P due to charge q: V = k₀(q/r), where r is the distance from the point charge.
• Absolute potential at infinity (at r=∞) is zero.

Electrical Potential Energy (PEE)

• To carry a charge q from infinity to a point where the absolute potential is V, work qV must be done on the charge.
• Work appears as electrical potential energy (PEE).
• When a charge q is carried through a potential difference V, work qV must be done on the charge.
• This work results in a charge qV in the PEE of the charge.
• For potential rise, V is positive and PEE increases if q is positive.
• For potential drop, V is negative, and the PEE decreases if q is positive.

V Related to E

• In a region with a uniform electric field E in the x-direction, the work done moving a test-charge through a distance x is W = Fx * x.
• V = Ex * x
• The field between two large, parallel, oppositely charged, closely spaced metal plates is uniform.
• The electric field E relates to the plate separation d and potential difference V as V = Ed.

Current

• A current I of electricity exists in a region when a net electric charge is transported from one point to another in that region.
• If a charge q is transported in a given cross section of the wire in a time t, then the current through the wire is I = q/t.
• q is in coulombs, t is in seconds, I is in amperes (1 A = 1 C/s).
• By custom, the direction of the current is the direction of flow of positive charge (flow of electrons to the right corresponds to a current to the left).
• A battery is a source of electrical energy
• If no internal energy losses occur, the potential difference between its terminals is the electromotive force (emf) of the battery.
• The terminal potential difference of a battery is equal to its emf unless otherwise stated.
• Unit for emf is the volt.

Resistance and Ohm's Law

• Electrical resistance of a circuit component/device is the ratio of voltage applied to the electric current: R = V/I
• Ohm's Law (I = V/R) can predict material behavior if resistance is constant over a range of voltage.
• It applies for both DC and AC applications.
• Resistance can be described via bulk resistivity.
• Resistivity and resistance are temperature dependent.
• Over sizable temperature ranges, temperature dependence can be predicted from a temperature coefficient of resistance.

Resistivity

• Electrical resistance is expected to be: greater for a longer wire, less for a wire of larger cross-sectional area, dependent upon the material
• Resistance of a wire is: R = ρl/A where ρ is the resistivity, l is the length, and A is the cross-sectional area
• Factor in the resistance which takes into account the nature of the material is the resistivity.
• Although it is temperature dependent, it can be used at a given temperature to calculate the resistance of a wire of given geometry.
• Inverse of resistivity is called conductivity.

Resistance Varies with Temperature

• If a conductor has resistance R₀ at temperature T₀, then resistance R at temperature T is: R = R₀ + αR₀(T - T₀).
• α is the temperature coefficient of the material of a conductor (usually varies with temperature, relation applicable over a small temperature range).

Circular Mil

• In engineering, unit of area of a round conductor is the circular mil (cmil).
• Mil is a unit of length equal to 0.001 in = 1/1000 in.
• Circular mil is a unit of area equal to the area of a circle whose diameter is 1 mil.
• Area Acm in circular mils of a circle whose diameter in mils is Dmil = D²mil
• The advantage of using the circular mil as a unit of area avoids multiplication and division by π.
• Given length of the conductor by feet/area in circular mils, the unit of resistivity is the ohm-cmil per foot.

Electric Power

• Rate at which work is done to maintain an electric current is given by the product of the current I and the potential difference V.
• P = IV where I is in amperes, V is in volts, and P is in watts.
• If the conductor/device obeys Ohm's Law, the power consumed may be expressed as: P = IV = I²R = V²/R.
• Electrical power delivered by an energy source as it carries charge through the potential rise in time: Power finished = Work/time.
• Since W = qV, P = W/t = qV/t

Resistors in Series

• Current is constant (magnitude of current through each resistor is equal).
• Total resistance is the algebraic sum of all resistances: RT = R1 + R2 + R3 + ... + Rn.
• Terminal voltage equals the algebraic sum of voltage drops: VT = V1 + V2 + V3 + ... + Vn.

Resistors in Parallel

• Voltage is constant (terminal voltage equals potential difference of each resistor).
• Reciprocal of resistance is called conductance.
• Total conductance is the algebraic sum of conductances: 1/R_T = 1/R_1 + 1/R_2 + 1/R_3 + ... + 1/R_n
• Total current equals the algebraic sum of branch currents: IT = I1 + I2 + I3 + ... + In

Resistors in Series-Parallel Circuit

• Most electrical circuits contain a combination of series and parallel circuits.
• Analyzing these circuits -> apply series/parallel techniques individually to to produce a much simpler overall circuit.