Electric Force and Charge

• Electric force is a fundamental force, focusing on electric charge, electric forces, properties of electrostatic forces, Coulomb's law, simple charge distributions, electric fields, and electric potential.

Electric Charge

• Electric charge (q) is a physical property causing matter to experience force near other charged matter.
• In atoms, electric charge occurs when the number of protons in the nucleus differs from the number of surrounding electrons.
• A body can be positively charged by losing electrons or negatively charged by gaining them.
• There are two types of charges: positive and negative.
• Like charges repel, while opposite charges attract.
• The SI unit of charge is the Coulomb (C).
• The smallest unit of free charge is the charge of an electron or proton, approximately e = 1.6 × 10⁻¹⁹ C.
• Electrons carry a negative charge (-e), and protons carry a positive charge (+e).
• Charge is quantized in integral multiples of e, which is a property called quantization of electric charge.
• The conservation of electric charge principle states that the net charge of an isolated system remains unchanged over time.
• It is possible to create or destroy charged particles, but not net charge.

Coulomb's Law

• Coulomb's Law quantifies the magnitude of electrostatic force, describing electrostatic interaction between charged particles.
• The law states: "The magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them."
• The magnitude of the electric force F between two charged particles with charges q1 and q2, separated by a distance r, is given by F = k * |q1q2| / r².
• Here, k is the Coulomb constant.
• The electrostatic constant (k) can be measured experimentally and is approximately 9 × 10⁹ Nm²/C².
• k is often expressed as k = 1 / (4πε₀), where ε₀ is the permittivity constant of free space, approximately 8.85 × 10⁻¹² C²/Nm².
• Coulomb's Law can be written as F = (1 / 4πε₀) * |q1q2| / r².
• Coulomb's equation calculates the magnitude of the force and requires a diagram to determine the direction.
• Opposite charges attract, while charges with the same sign repel.
• Coulomb's Law is valid for point charges, and the net force with multiple charges is the vector sum of all forces (superposition).

Electric Field

• An electric field exists in the space around any charged object, defined as a region around a source charge where other charged bodies experience an electric force.
• A charged particle propagates a "field" into all space, which other charged particles sense.
• Electric field is defined by the force it exerts on a small test charge q₀: Ẻ = F / q₀.
• The subscript "0" indicates the force is on the "test charge."
• The correct definition: Ẻ = lim (q₀→0) F / q₀.
• The equation F = qẺ calculates the force on a charged particle in an electric field.
• The direction of the electric field is the direction of the force on a positive test charge.
• Electric field units are newtons/coulomb (N/C) or volts/meter (V/m).
• The electric field exists independently of whether there is a charged particle present.
• The electric field direction is away from positive and towards negative charges.

The Electric Field Due to a Point Charge

• To determine the magnitude and direction of electric field intensity, consider a positive point charge q as a source charge and place a positive test charge q₀ at a distance r away at point P.
• The magnitude of the electric force exerted on q₀ from Coulomb's law is F = k * |q q₀| / r².
• The magnitude of electric field intensity at point P is E = F / q₀.
• If the point charge q is positive, then both the force F on the positive test charge q₀ and the electric field intensity Ē at point P are directed away from q.
• When the source charge q is negative, the force F and the electric field intensity E at point P are directed toward q.

Dipole

• A dipole is a combination of two electric charges with equal magnitude and opposite sign, separated by a fixed distance.
• The charge on the dipole is q (not zero, not +q, not -q, not 2q), and the distance between the charges is d.
• Dipoles are common and the electric field lines shown.

Motion of a Charged Particle in a Uniform Electric Field

• A charged particle in an electric field experiences a force; if free to move, acceleration is produced.
• If the only force is due to the electric field, then Σ F = mả = qẺ.
• If E is constant, acceleration a is constant, and kinematic equations are applicable.

Electric Potential, Potential Difference and Electrical Potential Energy

• A charged particle free to move in an electric field is analogous to a massive object free to move in a gravitational field, except massive objects primarily move in one direction in a gravitational field.
• An external force displaces an object uphill or the gravitational force displaces an object downhill in gravitational field.
• An external force displaces a charge in an electrical field.

Electric Potential and Potential Difference

• Electric potential (V) is the work needed to move a unit charge from a reference point to a specific point against an electric field, a scalar quantity.
• Electric potential = Work / Charge: V = W / q = U / q.
• The SI unit of electric potential is the Joule per Coulomb (J/C), or Volts (V). 1V = 1 J/C.
• Electric potential difference (p.d) or voltage (ΔV or V) between two points is the difference in electric potential between final and initial locations when work changes potential energy.
• ΔV = VB - VA = W / q = U / q
• A voltmeter measures electrical potential difference between two points in an electric circuit.

Electric Potential Energy

• Electric potential energy results from conservative Coulomb forces.
• Change in electric potential energy of the system is the negative of the work done by electric forces.
• This is also the negative of the work done by electrostatic force to bring a charged particle or the product of the charge and the electric potential: U = qV.
• Formula for two charges q1 and q2: U = k * (q1q2 / r).
• The electron volt (eV) is a unit of energy used, representing the energy change as an electron (or proton) moves through a potential difference of 1 Volt, where 1 eV = 1.6 × 10-19 C × 1 V = 1.6 × 10-19 J.

Electric Potential Due to a Point Charge

• To determine electric potential, consider a positive point charge q at point O as a source charge, and place a positive test charge q' at point P a distance r away.
• By definition, the electric potential at point P is V = U / q.
• Electric potential energy at point P equals the work to move a charge from O to P, given by W = U = -∫ F dr.
• If F = k * (qq'/r²), then U = -∫k(qq'/r²) dr = kqq'∫r⁻² dr = kqq' * r⁻¹
• Therefore the electric potential V = U/q' = kqq'/r / q' = kq/r.
• With multiple point charges, use the superposition principle to find the resulting potential: V = Σ(k qi / ri).

Potential Difference in a Uniform Electric Field

• Consider a charge q moving in a uniform electric field Ẻ.
• If the path is parallel to Ẻ, the potential difference between points A and B is ΔV = VB - VA = V = - ∫q Fds / q = - ∫qEqds / q = - ∫ Eds.
• ΔV = VB - VA = - ∫ Eds = -E ∫ ds = −Ed

Potential Gradient (Deriving Electric Field from the Electric Potential)

• E can be used to calculate V: V = VB - VA = - ∫ Eds.
• For two points separated by a small distance ds, dV = -Eds.
• The differential version calculates E from a known V: E = dV / ds.
• In one dimension: Ex = -dV/dx.
• In three dimensions: Ex = - ∂V/∂x, Ey = - ∂V/∂y, Ez = - ∂V/∂z.