Notes on Electric Fields and Electric Potential

Electric Field

• A region of space around an electrically charged object where a force would be exerted on other electrically charged objects.

• Analogized to a room with a strong scent, where you only perceive the scent when entering and having a functioning sense of smell.

• A perturbation of space caused by the presence of at least one electric charge.

• To detect the electric field of a charged body, another small charged body, called a test charge, is brought close. If the test charge experiences a force, then an electric field is present.

Electric Field Definition

• The presence of a charge distribution causes a force to be exerted on a charge $Q_0$ placed at a point P in space.

• It is the ratio of the force $F$ exerted on a positive test charge $Q<em>0$ to the charge $Q</em>0$: $E = \frac{F}{Q_0}$

• The electric field is a deformation of space produced by one or more charges.

• Measured in Newtons per Coulomb (N/C) in the International System of Units (SI).

Properties of the Electric Field

• Superposition Principle: The total electric field generated by a charge distribution $S$ at a point $P$ is the vector sum of the individual electric fields that each charge in $S$ generates at $P$.

• The electric field contains all the physical information needed to determine the electric force on a charge $Q$ placed at any point in space.

  • If q > 0, $F$ has the same direction and sense as $E$.

  • If q < 0, $F$ has the same direction but the opposite sense to $E$.

Electric Field of a Point Charge

• The electric field created at point $P$ by a point charge $Q$ located at $O$ is a vector.

  • Has a magnitude of $\frac{Q}{k r^2}$ (in vacuum).

  • Directed along the line $OP$.

  • The direction is outward from $O$ if $Q$ is positive and towards $O$ if $Q$ is negative.

Electric Field of a Point Charge (cont.)

• The simplest example of an electric field is that produced by a point charge $Q$ at the origin of a Cartesian coordinate system.

• The electric force between a point charge $Q$ and a test charge $q_0$ at a distance $r$ from the origin is given by Coulomb's law.

• The intensity of the electric field is $E = \frac{F}{q_0}$.

• If the charge $Q$ generating the field is positive, the field is oriented radially outward from the charge. If Q < 0, the field is oriented radially inward towards the negative charge.

Electric Field Lines

• A way to visualize the electric field qualitatively.

• A test charge $q$ is placed at a point $P<em>1$ in an electric field. Let $E</em>1$ be the field at $P_1$.

• The charge is moved in the direction and sense of the field from $P<em>1$ to a second point $P</em>2$. Let $E<em>2$ be the field at $P</em>2$.

• This operation is repeated, moving the test charge from $P<em>2$ to $P</em>3$ in the direction and sense of the electric field vector $E<em>2$. Let $E</em>3$ be the electric field vector at point $P_3$.

• By continuing this process, a broken line consisting of segments $P<em>1P</em>2$, $P<em>2P</em>3$, $P<em>3P</em>4$, $P<em>4P</em>5$, etc., is constructed, which identifies the direction and sense of the electric field at various points.

• If the displacements are very small, the broken line coincides with the line joining the various points $P<em>1$, $P</em>2$, $P_3$…

• A line constructed in this way is defined as a field line or line of force.

Electric Field Lines: Graphical Representation

• For each point in space, there is only one field line, oriented like the electric field at that point.

• Lines of force emerge from positive charges and enter negative charges.

• Lines of force are denser where the field is more intense.

• Field lines are only a useful representation of the field; they do not physically exist.

Examples of Electric Field Lines

• Electric field generated by a point charge.

• Electric field generated by two charges. In the case of charges of the same sign, the fields tend to cancel out in the central zone, and the field lines move radially to infinity (the test charge is repelled to the right by charge $q<em>1$ and to the left by charge $q</em>2$).

Uniform Electric Field: Parallel Plate Capacitor

• A special case of an electric field is that generated by two parallel plates, one charged positively and the other negatively.

• The field lines are parallel to each other, perpendicular to the two plates, and directed from the positively charged plate to the negatively charged plate.

• In the ideal case of infinitely large plates, the electric field is uniform (constant in magnitude, direction, and sense at every point) between the plates and zero outside.

• This is a good approximation for real plates if the distance between the two plates is much smaller than the dimensions of the plates themselves.

Electric Potential Energy

• Energy is the capacity of a body to do work.

• The electric field possesses energy because it has the capacity to do work.

• Consider a uniform electric field; we want to calculate the work done by the electric force to move a test charge from point A to point B.

• The electric force is, by definition, directly proportional to the electric field, and assuming the test charge is positive, the electric force and the electric field have the same direction and sense.

• The force vector is decomposed into its two components, one parallel and one perpendicular to the displacement vector.

• Since the work done by a force perpendicular to the displacement is zero, we have $L = F<em>\parallel d = q E</em>\parallel d$, where $E_\parallel$ represents the component of the electric field parallel to the displacement.

Electric Potential Energy (cont.)

• Since energy is defined as the capacity of a body to do work, the work done by the electric force to move a positive electric charge from point A to point B is defined as the difference between the electric potential energy of the charge at point A and the electric potential energy of the charge at point B:

• Therefore, the electric potential energy depends not only on the electric field but also on the charge $q$ being moved.

Potential Difference

• It is possible to define a new quantity that does not depend on the charge but only on its position in the electric field: the potential difference (d.d.p.).

• The d.d.p. measures the "electrical height difference" between two points in the field.

• Consider the electric field generated by a point charge $Q+$, in which we place a test charge $q+$. The latter moves from a point where the electrical level is high (closer to the charge that generated the field, where the field is more intense) to another point where the electrical level is lower.

  • Symmetrically, a negative test charge moves from a point where the electrical level is lower to another point where it is higher.

Potential Difference - Definition

• The potential difference between two points A and B of the electric field is defined as the ratio between the work done by the forces of the field to move the charge $q$ from point A to point B and the charge $q$ itself: $V = \frac{L}{q}$.

• The unit of measurement for potential difference in the SI is J/C. The ratio J/C is a new unit of measurement called Volt (V) in honor of the Italian physicist Alessandro Volta, inventor of the battery.

Potential Difference and Charge Motion

• From the definition of d.d.p., we derive the inverse formula that gives us the work done by the forces of the field to move a charged particle between any two points A and B:

• $L = q \cdot V$

• From this relation, we can derive predictions about the motion of charged particles.

• If a charged particle $q$ is set in motion by the forces of the electric field (F and s have the same direction), then the work done by them is positive.

Potential Difference and Charge Motion (cont.)

• The product is positive if:

  • 1. q > 0 and V > 0

  • 2. q < 0 and V < 0

• Therefore:

  • Particles with positive electric charge, under the action of the forces of the electric field, move from points of higher potential to points of lower potential.

  • Particles with negative charge, under the action of the forces of the electric field, move from points of lower potential to points of higher potential.

Key Points on Potential Difference

• Potential difference is measured in volts and is indicated by ΔV.

• A positive charge moves spontaneously (L>0) towards the lower potential: Q>0; L>0 => VA > VB

• A negative charge moves spontaneously (L>0) towards the higher potential: Q<0; L>0 => VA < VB

Relationship Between Electric Field and d.d.p.

• Consider two charged plates, of opposite sign and very small distance d compared to their surface area. The electric field generated between them is uniform and perpendicular to the two plates.

• If the field is uniform, there is a very simple relationship between field and potential: $ΔV = E \cdot d$

• The intensity of the electric field inside a capacitor whose plates are at a distance d and have a potential difference ΔV is given by:

• $E = \frac{ΔV}{d}$. The intensity of the electric field can therefore be measured in Volts/meter (V/m).

Practical Implications

• When we talk about a 12 Volt battery, we mean that the battery generates an electric potential difference of 12V.

• The potential difference is often called voltage.

Capacitors

• Capacitors are devices that store electrical energy and charge.

• They are used in electrical systems to accumulate energy.

• A capacitor consists of two conductors (armatures) separated by a distance. These plates acquire charges of equal magnitude but opposite sign (+Q and -Q) when connected to a battery with potential difference AV.

• The charge Q is directly proportional to the potential difference AV: Q=CAV

Capacitance

• Capacitance (C) is the ratio of the charge (Q) on the plates to the potential difference (AV) between them.

• $C = \frac{Q}{AV}$

• Measured in Farads (F), where 1F = 1C/1V.

Capacitor - Water Bucket Analogy

• The properties of a capacitor can be compared to a water bucket.

  • The cross-sectional area of the bucket represents the capacitance C.

  • The amount of water in the bucket represents the charge Q.

  • The height of the water in the bucket represents the potential difference AV.

• A wider bucket holds more water at the same height, just as a capacitor with larger C can store more charge at the same AV.

Interdisciplinary Connections

• Modern society relies heavily on electrical appliances, a development primarily from the 20th century's Second Industrial Revolution.

• A key discovery was electric current, produced when charged particles move orderly through established and closed paths.

• This kinetic energy is converted into other forms, powering appliances like vacuum cleaners or fans.

Electric Current

• Electric current is the flow of electric charge through a conductor per unit of time.

• $I = \frac{ΔQ}{Δt}$. Measured in Amperes (A), where 1A = 1C/1s.

Electric Current Analogy

• In fluid conductors, a height difference yields fluid flow, which lessens the height difference. When levels are equal, flow stops. To keep flow, a pump recreates the height difference.

• In electric circuits, potential difference yields electric current, which reduces the potential difference. Continuous current requires a voltage source (like a battery).

• Batteries act like pumps, lifting positive charges to higher potential.

• Generators of continuous voltage maintain a constant potential difference between terminals.

Moving Charges in a Conductor

• To move charges within a conductor, a potential difference must exist.

• Positive charges move toward lower potential (in the direction of the electric field), while negative charges move toward higher potential (opposite the electric field).

• Conventional current direction is defined as the direction of positive charge flow, even in metal conductors where electrons (negative charges) are moving.

• Mathematically, a flow of negative charges to the left equals a flow of positive charges to the right.

Electric Current - Charge Details

• The charge ΔQ in the current intensity formula is the sum of positive charges crossing a section S of the conductor in the conventional direction and negative charges crossing S in the opposite direction.

Electric Circuits

• An electric circuit is a set of conductors connected continuously and to a generator.

  • DC CIRCUITS: Current flows in one direction.

  • AC CIRCUITS: Current periodically reverses direction.

Electric Circuits (cont.)

• When a voltage generator's terminals are linked via conductors, a continuous loop forms with two phases:

  • Current flows outside of the generator from positive to negative pole

  • The generator transports internal charges from negative to positive pole.

• The generator and circuit share the same current.

Electromotive Force (EMF)

• When a battery is disconnected and not generating current, the potential difference between poles is the electromotive force (EMF), measured in Volts (V).

• EMF is not a force but the work done to transport a charge along the circuit. The battery maintains a potential difference between points.

• With a closed circuit, potential difference is slightly less than EMF because some energy is used to move internal charges.

Ohm's First Law

• Current (i) in a metal wire is proportional to the potential difference (ΔV) across it.

• $\Delta V = R i$ where R is the resistance.

• The unit of resistance is the ohm (Ω): $\Omega = \frac{V}{A}$

Resistance vs. Resistor

• Resistance: A physical property of a material.

• Resistor: A component of a circuit.

Ohm's Second Law

• The resistance R of a wire with length L and cross-sectional area A is:

• $R = ρ \frac{L}{A}$, where ρ is the material's resistivity, depending on the material and temperature.

Resistors in Series and Parallel

• Series Connection: Same current through all resistors.

• Parallel Connection: Same potential difference across all resistors.

• Series example: Christmas lights. When one bulb fails, the circuit breaks, and all lights go out.

• Parallel example: Home electrical systems. Devices operate independently with the same potential difference.

Resistors in Series (Detailed)

• The current $i$ is the same through all resistors.

• The total potential difference is the sum of individual differences: $ΔV = ΔV<em>1 + ΔV</em>2$

• Applying Ohm's law at each resistor: $ΔV<em>1 = R</em>1 i$ and $ΔV<em>2 = R</em>2 i$

• Therefore:

  • $ΔV = R<em>1 i + R</em>2 i = (R<em>1 + R</em>2) i$

  • $R<em>{eq} = R</em>1 + R_2$

• The equivalent resistance of n series resistors is the sum of individual resistances:

• $R<em>{eq} = R</em>1 + R<em>2 + … + R</em>n$

Resistors in Parallel (Detailed)

• The current from the generator equals the sum of currents through each resistor: $i = i<em>1 + i</em>2$

• Resistors have the same potential difference: $i<em>1 = \frac{ΔV}{R</em>1}$, $i<em>2 = \frac{ΔV}{R</em>2}$

• Therefore:

  • $\frac{1}{R<em>{eq}} = \frac{i}{ΔV} = \frac{1}{R</em>1} + \frac{1}{R_2}$

Resistors in Parallel - cont.

• The inverse of the equivalent resistance $R_{eq}$ of multiple resistors in parallel is the sum of the inverses of the individual resistances:

• $\frac{1}{R<em>{eq}} = \frac{1}{R</em>1} + \frac{1}{R<em>2} + \frac{1}{R</em>3} + …$

• Adding parallel resistances is like opening more supermarket checkout lanes: Adding a resistor in parallel provides an extra channel for current, decreasing overall resistance.

Capacitors in Series and Parallel

• Series Connection: One capacitor's plate is connected to the other's plate.

• Parallel Connection: Capacitors have the same potential difference.

Capacitors in Series and Parallel - Formulas

• The inverse of the equivalent capacitance $C_{eq}$
  of multiple capacitors in series is the sum of the inverses of the individual capacitances:

• $\frac{1}{C<em>{eq}} = \frac{1}{C</em>1} + \frac{1}{C<em>2} + \frac{1}{C</em>3} + …$

• The equivalent capacitance $C_{eq}$ of multiple capacitors in parallel is the sum of the individual capacitances:

• $C<em>{eq} = C</em>1 + C<em>2 + C</em>3 + …$

Capacitors in Parallel - PROOF

• Let $Q<em>1$ and $Q</em>2$ be the charges accumulated on capacitors $C<em>1$ and $C</em>2$, respectively. There is the same potential difference between the plates of each capacitor because the capacitors are connected in parallel.

• $Q<em>{tot} = Q</em>1 + Q_2$

• From the definition of capacitance, we obtain $Q = C*ΔV$

• $C<em>{eq}ΔV= C1</em>ΔV + C<em>2*ΔV$ dividing both sides by $ΔV$ we obtain $C</em>{eq}=C<em>1 + C</em>2$

Capacitors in Series - PROOF

• The same charge is present on the plates of capacitors in series (because they are crossed by the same current intensity) while $ΔV=ΔV<em>1+ΔV</em>2 (*)$

• We know that the capacity of a capacitor is defined as the ratio between the amount of charge on the plates and the potential difference between the plates $C=Q/ΔV$ --> $ΔV=Q/C$

• So substituting in (*) we get: $Q/C<em>{eq}=Q/C</em>1 + Q/C_2$; we divide both sides by Q by applying the second principle of equivalence

• $1/C<em>{eq}=1/C</em>1 + 1/C_2$