Physics

Electric Field

Electric charge, Coulomb's law, electric field, Gauss's law, electric potential and energy.

The electron and the proton are the charged particles we find as components of atoms. Their charges are exactly equal in magnitude but opposite in sign. The charge of a proton is +e+e and that of an electron is −e-e, where ee is the elementary charge.
The SI unit of electric charge is the coulomb (C). The fundamental unit of electric charge e is:
e=1.602⋅10−19 Ce = 1.602 \cdot 10^{-19} \text{ C}
The electrostatic force between two point charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them, and acts along the line joining them. The force is repulsive if the charges have the same sign, and attractive if they have opposite signs.
Electrostatic force between two charges
F⃗12=KQ1⋅Q2r2u⃗12(K=14πε)\vec{F}_{12} = K \frac{Q_1 \cdot Q_2}{r^2} \vec{u}_{12} \quad \left( K = \frac{1}{4\pi\varepsilon} \right)
Where F⃗12\vec{F}_{12} is the force exerted by charge 1 on charge 2, u⃗12\vec{u}_{12} is the unit vector pointing from charge 1 to 2, K is a constant that depends on the medium, and ε\varepsilon is the electric permittivity of the medium, which indicates its ability to polarize (align its particles with the electric field).
Its value in a vacuum (or air) is ε0=8.85⋅10−12C2N⋅m2\varepsilon_0 = 8.85 \cdot 10^{-12} \frac{\text{C}^2}{\text{N} \cdot \text{m}^2}, therefore the proportionality constant is:
K=9⋅109N⋅m2C2K = 9 \cdot 10^9 \frac{\text{N} \cdot \text{m}^2}{\text{C}^2}
The electric field E is a property of the space around an electrically charged particle. The electric field, or electric field strength, at a point is defined as the net force per unit of positive test charge.
E⃗=F⃗qorE⃗=KQr2u⃗\vec{E} = \frac{\vec{F}}{q} \quad \text{or} \quad \vec{E} = K \frac{Q}{r^2} \vec{u}
Where u is the unit vector from the source charge to the point in space.
The superposition principle applies: if there are several charges influencing a point in space, the total electric field at that point is the vector sum of the individual electric fields produced by each charge:
E⃗T=E⃗1+E⃗2+…\vec{E}_T = \vec{E}_1 + \vec{E}_2 + \dots
Electric field lines. They indicate the direction of the force that a positive test charge would experience at each point in the field.
Electric field lines
The electric flux is defined as the measure of the electric field passing through a surface: Φ=E⃗⋅S⃗\Phi = \vec{E} \cdot \vec{S}
If the surface is not flat or the field is non-uniform, the flux must be calculated as an integral over infinitesimal area elements dS⃗d\vec{S}:
Φ=∫E⃗⋅dS⃗\Phi = \int \vec{E} \cdot d\vec{S}
Gauss's Law: The total electric flux through a closed surface is directly proportional to the enclosed electric charge:
Φ=∮E⃗⋅dS⃗=Qencε\Phi = \oint \vec{E} \cdot d\vec{S} = \frac{Q_{enc}}{\varepsilon}
Electric Potential Energy
The electric field is conservative. The work required to move a charge between two points does not depend on the path taken, only on the initial and final positions.
  • The work done to move a charge qq around a closed loop is zero.
Work done by the electric field: The work done by the field created by Q on a charge qq moving between two points is:
WA→B=∫rArBF⋅dr=∫rArBKQ⋅qr2dr=K⋅Q⋅q(1rA−1rB)W_{A \to B} = \int_{r_A}^{r_B} F \cdot dr = \int_{r_A}^{r_B} K \frac{Q \cdot q}{r^2} dr = K \cdot Q \cdot q \left( \frac{1}{r_A} - \frac{1}{r_B} \right)
Electric potential energy (UeU_e) at point A is the work done by the field to move a charge qq from A to infinity.
Ue=KQ⋅qrA(J)U_e = K \frac{Q \cdot q}{r_A} \quad \text{(J)}
Electric Potential
It is the electric potential energy per unit charge:
V=UeqorV=KQr(volts V = J / C)V = \frac{U_e}{q} \quad \text{or} \quad V = K \frac{Q}{r} \quad \text{(volts V = J / C)}
Potential difference: The potential difference between two points can be determined if the electric field E is known:
ΔV=VB−VA=−∫rArBE⃗⋅dr⃗\Delta V = V_B - V_A = -\int_{r_A}^{r_B} \vec{E} \cdot d\vec{r}
Superposition principle: The electric potential in a region influenced by multiple point charges is the algebraic sum of the individual potentials.
VT=V1+V2+V3+…V_T = V_1 + V_2 + V_3 + \dots
Work done by an electric field to move a charge from point A to point B is the negative change in potential energy between those points.
WA→B=−ΔUe=q(VA−VB)W_{A \to B} = -\Delta U_e = q(V_A - V_B)
Kinetic energy acquired by a charge moving through an electrostatic potential region results from the transformation of electric potential energy into kinetic energy (K):
K=q(−ΔV)=q(VA−VB)orm⋅v22=q(VA−VB)K = q(-\Delta V) = q(V_A - V_B) \quad \text{or} \quad \frac{m \cdot v^2}{2} = q(V_A - V_B)