Electric Charge

Configuration of two point charges:

• A source charge $Q$ (Q > 0) is located at a fixed position in space.

• A test charge $q$ is located at a point $P$ in space at a distance $r$ from $Q$.

• The unit vector \normalfont{\text{\bfseries r}} (also referred to as $\frac{\bf r}{r}$) points directionally from charge $Q$ to test charge $q$.

• With like charges (both positive), the charges experience a repulsive force.

• With unlike charges (one charge negative), the charges experience an attractive force.

Coulomb's Law:

• The electrostatic force $\bf F$ acting on test charge $q$ is mathematically described by: \textbf{F} = k\textsubscript{c} \times \frac{q \times Q}{r^2} \textbf{\textit{r}}

• k\textsubscript{c} represents the Coulomb constant.

• Same sign leads to a force acting along \normalfont{\text{\bfseries \textit{r}}} outward.

• Opposite sign leads to a force acting against \normalfont{\text{\bfseries \textit{r}}} inward.

Definition of Electric Field (E-field):

• The electric field at point $P$ is represented by vector $\bf E$.

• Formally, the electric field corresponds to the electrostatic force per unit test charge: $\textbf{E}(P) = \frac{\textbf{F}}{q}$.

• Division fully eliminates the test charge $q$: \textbf{E}(P) = k\textsubscript{c} \times \frac{Q}{r^2} \textbf{\textit{r}}.

• The electric field $\bf E$ is independent of the test charge $q$ and represents the property of space created by charge $Q$ (or any arbitrarily complicated charge configuration).

Direction Convention of the E-field:

• The electric field $\bf E$ always points in the direction that a positive test charge would experience a force.

• If Q > 0, $\bf E$ points radially outward from $Q$.

• If Q < 0, $\bf E$ points radially toward $Q$.

Units of the Electric Field:

• In the SI unit system, the unit is Newton per Coulomb: $\text{N}\text{C}^{-1}$ (or $\text{N/C}$).

Superposition Principle and Force Calculation:

• When multiple source charges q\textsubscript{1}, q\textsubscript{2}, q\textsubscript{3}, \text{...}, q\textsubscript{i} are present, the resulting electric field at point $P$ is determined by the vector addition of the individual field contributions: \textbf{E}_{\text{net}}(P) = \textbf{E}_1 + \textbf{E}_2 + \textbf{E}_3 + \text{...} + \textbf{E}_i = \text{∑}_{i} \textbf{E}_i.

• Validity: The superposition principle is not trivial or obvious to derive but is considered a fundamental postulate of electrostatics, as it perfectly matches all experimental observations.

Calculating Force on Any Charge in the E-field:

• If the electric field $\bf E$ is known at a location, the electrostatic force $\bf F$ on any charge $q$ placed there can be directly calculated: $\textbf{F} = q \times \textbf{E}$.

• For positive $q$, the force $\bf F$ acts parallel to $\bf E$.

• For negative $q$, the force $\bf F$ acts antiparallel to $\bf E$.

• The strength of the force scales directly proportional to the magnitude of $q$.

Quantitative Analysis of Charge Configurations and Null Points:

• System with a $+3$ charge and a $-1$ charge:

  • A configuration of two charges with relative values $+3$ and $-1$ at a fixed distance is considered.

  • Behavior near the $-1$ charge: Due to the $1/r^2$ decay, the influence of the $-1$ charge dominates locally. A very close positive test charge will be attracted to the $-1$ charge.

  • Behavior far away ($r \text{≫}$ charge distance): From a large distance, the system behaves like an effective total point charge of $+3 + (-1) = +2$.

  • The far field behaves like a $1/r^2$ field of a $+2$ charge, hence the force on a positive test charge there is strictly outward (away from the configuration).

• Existence of a Null Point (E-field = 0):

  • As the force on a positive test charge near the $-1$ charge points inwards, while far away it points outward, there must be a point on the line connecting them (on the side of the $-1$ charge) where the net E-field exactly vanishes: $\textbf{E}_{\text{net}} = 0$.

  • At this point, the repulsive force of the $+3$ charge and the attractive force of the $-1$ charge on a test charge perfectly cancel out.

Representation Forms: Field Vectors vs. Field Lines and Particle Dynamics:

• Representation by Field Vectors:

  • Vector arrows are drawn at discrete points.

  • Arrow direction indicates the force direction on a positive test charge.

  • Arrow length shows local field strength (gets shorter according to $1/r^2$ with increasing distance).

• Representation by Field Lines:

  • Field lines are continuous curves in space.

  • The tangent to a field line at any point gives the direction of the E-field (and thus the force direction on a positive charge).

  • Negative charges experience forces opposite the direction of the field lines.

  • Line Density: The density of field lines represents field strength (high density = strong field; low density = weak field).

  • Number of Lines: The number of lines emanating from or entering a charge is proportional to its charge magnitude (e.g., three times as many lines emanate from a $+3$ charge compared to those entering a $-1$ charge).

  • Analogy: Positively charged objects act like fans (blowing lines outward), negatively charged objects act like vacuums (sucking lines inward).

• Difference between Field Lines and Particle Trajectories:

  • Straight Field Lines: If a charge is released from rest ($v = 0$) in a straight field, it accelerates along the field line and remains exactly on it.

  • Curved Field Lines: If a charge is released from rest in a curved field, it experiences initial acceleration tangential to the field line. Once the charge has speed v > 0, inertia causes it to leave the curved field line.

  • Conclusion: Field lines generally do not represent the trajectories of charged particles.

Special Configurations: Maxwell Diagrams and Electric Dipoles:

• Like Charge Ratios (e.g., $1:4$ per Maxwell):

  • Consideration of two positive charges $+4$ and $+1$.

  • Both act as fans; between the charges, the repulsive forces cancel at a point near the smaller charge ($+1$) ($\textbf{E} = 0$).

  • Comparable to the gravitational null point in the Earth-Moon system.

• The Electric Dipole:

  • An electric dipole consists of two equal magnitude charges of opposite sign ($+q$ and $-q$).

  • Near Field: Near the positive charge, the lines radiate outward; near the negative charge, they enter radially.

  • Far Field ($r \text{≫}$ dipole distance): The total net charge is $+1 + (-1) = 0$. In the far field, neither charge predominates.

  • Distance Law of Dipole Field: The electric field of a dipole decreases at large distances proportional to $\frac{1}{r^3}$ (faster than the $1/r^2$ field of a point charge).

  • Null Points: There isn't a single point in the entire space where the E-field of a dipole is exactly $0$ ($\textbf{E} \neq 0$ everywhere in finite space).

Induced Dipoles and Experimental Demonstrations of Polarity:

• Dipole Induction in Atoms and Molecules:

  • When a spherical neutral atom or molecule is placed in an external electric field, the electrons are displaced against the E-field, while the positive nucleus is pressed in the field direction.

  • Over time, charge separation creates an induced dipole.

• Demonstration 1: Creation of a Dipole with Metal Balls and Detection Using an Electroscope:

  • Two touching conductive metal balls (free electrons) are initially neutral.

  • A negatively charged rubber rod (charged by rubbing) is brought near the connected balls.

  • Induction: Free electrons are pushed into the far ball (negative excess). The approaching ball receives an equal positive charge excess (charge conservation).

  • The balls are separated while the rubber rod is still present.

  • Result: Two separated, oppositely charged balls form a permanent dipole.

  • Detection with the Aluminum Strip Electroscope: - Contacting the electroscope with the negative ball leads to charge transfer; the aluminum strip deflects due to electrostatic repulsion.

    • Approaching (without contact) with the positive ball pulls electrons from the strip upwards \n → The electroscope's deflection decreases.\n - Approaching with a negative object pushes more electrons down  $→$ The deflection increases.

• Demonstration 2: Torque on a Dipole in an E-field and Alignment:

  • A dipole consists of two conductive table tennis balls (yellow marking $A$, orange marking $B$), mounted on an insulating stick.

  • In the external field, the positive side experiences a force in the field direction, the negative side against the field direction.

  • A torque $\boldsymbol{\tau}$ acts to turn the dipole clockwise/counterclockwise until it aligns parallel to the E-field lines.

  • Damping causes the dipole to come to rest after oscillations along the field lines.

• Demonstration 3: Probing the Radial E-field of a Van-de-Graaff Generator:

  • A dipole is induced using a temporary metal connector in the field of a Van-de-Graaff generator and carried on an insulating thread attached to a fishing pole.

  • When this permanent dipole is moved around a second Van-de-Graaff generator, it consistently aligns radially.

  • Polarity Analysis: If the negative part of the dipole (yellow) points outward and the positive inward, the two Van-de-Graaff generators used have opposite polarities.

Visual Field Line Representation and Interaction with the Human Body:

• Demonstration 4: Visualization of E-field Lines Using Grass Seeds in Oil:

  • Long grass seeds are suspended in a bowl of oil.

  • Applying an electric field polarizes the grass seeds by induction (electrons shift to one end).

  • The resulting torque rotates the grass seeds parallel to the local E-field lines.

  • Observed Patterns: - Dipole (plus/minus): Radial lines near the poles and continuous, curved field arcs between the two charges.

    • Like Charges (plus/plus): Lines bend away from each other; clear visualization of the mutual repulsion effect ("fan effect").

• Demonstration 5: Charged Person and Oscillating Balloon:

  • A person (Walter Lewin) stands grounded near a positively charged Van-de-Graaff generator.

  • By induction, electrons are drawn from the earth into the body; the person becomes effectively negatively charged.

  • A lightly conductive charged balloon is placed between the generator ($+$) and the person's head ($-$).

  • Movement Sequence: 1. The positively charged balloon is repelled from the generator and moves tangentially to the field lines toward the person's head.

    2. Upon contact with the head, the balloon gains negative charge.

    3. Due to the polarity reversal, the balloon is now repelled from the head and flies back to the generator.

    4. At the generator, it discharges, recharges positively, and flies back to the head.

  • The balloon continuously bounces between the person and the generator.