Electric Charges and Fields

The One Thing

Electric charges create electric fields, and Coulomb’s law determines the forces between them. Gauss’s law uses electric flux and symmetry to calculate electric fields for highly symmetric charge distributions.

Reactions, Processes and Experiments

Key Terms

• Electric charge: A fundamental property of matter responsible for electrical interactions; it may be positive or negative and is measured in coulombs.
• Quantisation of charge: Charge exists in discrete amounts given by $q=ne$, where $n$ is an integer and $e=1.6\times 10^{-19}\mathrm{C}$.
• Conservation of charge: The total electric charge of an isolated system remains constant; charge can be transferred but cannot be created or destroyed.
• Coulomb’s law: The electrostatic force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
• Superposition principle: The net force or electric field due to several charges is the vector sum of the forces or fields produced by each charge separately.
• Electric field: The force experienced per unit positive test charge placed at a point: $E=F/q_0$.
• Electric field due to a point charge: The field at distance $r$ from charge $q$ is $E=(1/4\pi {\epsilon }_0)(q/r^2)$, directed away from a positive charge and toward a negative charge.
• Electric field lines: Imaginary lines whose tangent gives the direction of the electric field; their density represents field strength, and they never intersect.
• Electric dipole: A pair of equal and opposite charges separated by a small distance.
• Electric dipole moment: A vector quantity defined as $p=qd$, directed from the negative charge to the positive charge.
• Torque on a dipole: A dipole in a uniform electric field experiences torque $\tau =pE\sin \theta$, tending to align it with the field.
• Electric flux: A measure of the electric field passing through a surface; for a uniform field, $\Phi =EA\cos \theta$.
• Gauss’s law: The total electric flux through a closed surface equals the enclosed charge divided by ${\epsilon }_0$: $\Phi =q_{\text{enclosed}}/{\epsilon }_0$.
• Gaussian surface: An imaginary closed surface selected according to symmetry to apply Gauss’s law conveniently.
• Electric field due to an infinite line charge: For linear charge density $\lambda$, the field at distance $r$ is $E=\lambda /(2\pi {\epsilon }_{0}r)$, directed radially outward for positive charge.
• Electric field due to an infinite plane sheet: For surface charge density $\sigma$, the field is $E=\sigma /(2{\epsilon }_0)$, independent of distance from the sheet.
• Electric field due to a uniformly charged spherical shell: Outside the shell, the field is the same as that of a point charge at the centre; inside the shell, the electric field is zero.
• Permittivity of free space: The constant ${\epsilon }_0$, approximately $8.85\times 10^{-12}{\mathrm{C}}^2{\mathrm{N}}^{-1}{\mathrm{m}}^{-2}$, that appears in electrostatic laws.

Easily Confused

• Electric field and electric potential: Electric field is a vector quantity, whereas electric potential is a scalar quantity; field calculations require vector addition.
• Electric flux and electric field: Electric field describes force per unit positive test charge, whereas electric flux measures the electric field passing through a surface.
• Point-charge, line-charge and plane-sheet fields: Their distance dependences are respectively $1/r^2$, $1/r$, and constant with distance.
• Uniformly charged spherical shell and uniformly charged solid sphere: The field inside a spherical shell is zero, whereas inside a uniformly charged solid sphere it is $E=(1/4\pi {\epsilon }_0)Qr/R^3$ for $r<R$.
• Field on and near a conductor: Excess charge resides on the conductor’s surface, the field inside is zero, and the field just outside is $E=\sigma /{\epsilon }_0$.
• Local electric field and net flux: Charges outside a closed Gaussian surface can produce field on the surface, but they make zero net contribution to the total flux.

What Gets Asked

• State or apply Coulomb’s law, including the inverse-square dependence, the constant $1/(4\pi {\epsilon }_0)=9\times 10^9\mathrm{N}{\mathrm{m}}^2{\mathrm{C}}^{-2}$, and the fact that the force acts along the line joining the charges. A common mark-losing slip is treating like charges as attracting or unlike charges as repelling.
• Calculate the resultant force or electric field from several charges using the superposition principle. The key error is adding magnitudes without performing vector addition.
• Define electric field, calculate $E$, or determine the force using $F=qE$. The field direction must be interpreted for a positive or negative charge.
• Calculate electric flux through an open or closed surface using $\Phi =EA\cos \theta$, $\Phi =\int E\cdot dA$, or $\Phi =\oint E\cdot dA$. The area vector of a closed surface is directed outward.
• Apply Gauss’s law to spherical, cylindrical, or planar symmetry. The Gaussian surface must match the symmetry so that the field is constant in magnitude or has a simple direction over the relevant surface.
• Compare fields for a point charge, an infinite line charge, an infinite plane sheet, a uniformly charged spherical shell, or a uniformly charged solid sphere. The main distinction is the correct distance dependence and the different inside and outside expressions.