Electric Fields and Capacitors Notes

Electric Charges and Forces

• Charge comes in two types: positive (+) and negative (-).

• SI Unit: Coulomb (C).

• Charges exert a force on one another:

  • Same signs repel.

  • Opposite signs attract.

Origins of Charge

• Electrons: negative.

• Protons: positive.

• Neutrons: neutral.

• Ionization: electron is pulled from or added to an atom, resulting in positive or negative ions.

Basic Properties of Electric Charge

• Electric charge is conserved.

  • It cannot be created or destroyed, only moved.

  • In reactions, the net charge must remain constant.

• Electric charge is quantized.

  • Electric charge comes in integer multiples of the electron charge $(e = -1.6 × 10^{-19} C)$.

  • The proton has the same charge as the electron but with opposite sign: $(+1.6 × 10^{-19} C)$.

Conductors and Insulators

• Conductors:

  • Have some electrons that can move freely around the entire material.

  • These are not tied to a particular nucleus.

  • This motion results in electric currents.

• Insulators:

  • All the electrons are tightly bound to specific atoms and cannot travel far from the nucleus.

  • These materials do not conduct electric currents.

Charge Transfer

• Friction: Triboelectric effect involves the transfer of charge through rubbing.

• Conduction: Direct contact allows electrons to move between objects.

• Induction: Charging an object without direct contact by bringing a charged object nearby, which polarizes the neutral object and allows charge to be transferred if grounded.

Polarization

• Most neutral atoms or molecules are not symmetric.

• Even though the net charge is zero, if you get close enough, you can feel the effect of their electric charges.

• Molecules align due to electrostatic attraction.

Electrostatic Forces in Molecular Biology

• Electrostatic forces play a crucial role in molecular interactions, such as hydrogen bonding in DNA base pairs.

Quantifying the Force: Coulomb’s Law

• Proportional to the product of the two charges.

• Inversely proportional to the square of the distance: $\frac{Q<em>1 Q</em>2}{r^2}$.

• $k$ is a constant, in SI units, $k = 9.0 × 10^9 Nm^2/C^2$. Also, $k = \frac{1}{4\pi\epsilon<em>0}$, $\epsilon</em>0 = 8.85 × 10^{-12} C^2/N ⋅ m^2$. $\epsilon_0$ is the permittivity of free space.

• Coulomb's Law: $F<em>c = k \frac{|Q</em>1Q_2|}{r^2}$

Force is a Vector

• The Coulomb force has a magnitude and direction (like all forces).

• The direction is along the line connecting the two charges.

• $\vec{F}_{12}$ is the force on charge 1 due to charge 2.

• Each charge in a pair feels a force from the other one; the two forces have the same magnitude and opposite direction (Newton’s third law).

• $|F<em>{12}| = |F</em>{21}| = k \frac{q<em>1q</em>2}{r^2}$

Superposition

• The net force on a charge due to multiple other charges is the vector sum of the individual forces.

• Example: Calculating the net force on charge $q$ due to charges $Q<em>1$ and $Q</em>2$.

  • $F_{net} = -0.37 × 10^9 N - 6.75 × 10^9 N = -7.12 × 10^9 N$

  • The minus sign indicates the net force is to the left.

Electric Fields

• The force between two charges occurs "at a distance": one charge can push or pull another one without "touching it."

• Each charge affects the space around it, creating an 'electric field'.

• The electric field maps the magnitude and direction of the force that would be experienced by a unit positive charge.

• SI Units: Newton/Coulomb (N/C).

• Electric field: $E = \frac{F}{q}$, where $F$ is the force on a test charge $q$.

Electric Field Lines

• Electric field lines provide a visual representation of the electric field.

• The density of field lines indicates the strength of the field.

• Field lines point away from positive charges and towards negative charges.

Superposition of Electric Fields

• The net electric field at a point is the vector sum of the electric fields due to all individual charges.

Cell Membrane - Separation of Charges

• The cell membrane maintains a separation of charges, leading to an electric potential difference.

• Ions such as $Na^+$, $Cl^-$, and $K^+$ are involved in maintaining this separation.

• Coulomb forces and diffusion processes play a role in establishing the charge distribution.

Electrical Potential Energy

• $\Delta U = \Delta \text{Potential Energy} = -W<em>{elec}$, where $W</em>{elec}$ is the work done.

• The change in potential energy is the negative of the work done by the electric force.

• For a uniform electric field, the potential energy change is $\Delta U = -qEd$, where $d$ is the distance moved in the direction of the field.

Work and Potential Energy

• The electrical potential energy changes when a charge moves in an electric field.

• The work done by the electric field is related to the change in potential energy.

Electric Potential

• Electric potential $V$ is potential energy per unit charge at some point A.

• Only a difference in potential energy between two points A and B is physically meaningful.

• Potential Difference: $\Delta V = V<em>B - V</em>A = \frac{PE<em>B - PE</em>A}{q}$.

Electric Potential vs Potential Energy

• Both are scalar quantities (not vectors) but still have a sign.

• Potential energy Unit: Joule, J.

• Electric potential Unit: Volt, V = J/C.

• Only differences in potential energy or in electric potential are meaningful.

• Electric potential difference is also called voltage.

Electric Potential vs Electric Field

• For a parallel plate capacitor where $E$ is constant:

• Electrical potential difference: $V_{BA} = \frac{\Delta PE}{q} = -Ed$.

• Electric field is then $E = \frac{V_{ba}}{d}$.

• $\Delta PE = PE<em>B - PE</em>A = -W = -Eqd$

Equipotential Lines

• Lines/surfaces of constant potential. No work is done when moving a charge along these lines.

• Always perpendicular to the electric field lines.

Electric Potential of a Point Charge

• We choose $V = 0$ at $r = \infty$.

• The equipotentials are spheres centered on the charge; notice how the electric fields are perpendicular to them.

• For a point charge: $V(r) = -Er = k \frac{Q}{r}$.

Capacitors

• A capacitor is a device to store energy by separating charges.

Capacitance

• In a parallel-plate capacitor connected to a battery, the charge is proportional to the voltage of the battery.

• $C$ is the capacitance.

• SI Unit of C: Farad (F). 1F = 1C/V.

• For a parallel plate capacitor: $C = \epsilon_0 \frac{A}{d}$.

• $\epsilon_0 = 8.85 × 10^{-12}F/m$ (permittivity of free space).

Dielectrics

• Dielectrics are insulators whose molecules tend to orientate to reduce the external field.

• For a parallel plate capacitor with a dielectric: $C = \kappa \epsilon_0 \frac{A}{d}$.

• $\kappa$ is the dielectric constant.

Storage of Electric Energy

• Energy stored = work done to separate charges:

• $U = \frac{1}{2}QV = \frac{1}{2}CV^2 = \frac{1}{2} \frac{Q^2}{C}$.

• The energy density, defined as the energy per unit volume, can be written in terms of the electric field:

• $\text{Energy density} = \frac{U}{\text{Volume}} = \frac{1}{2} \epsilon_0 E^2$