Unit 8: Magnetism, Electrostatics, and Circuit Analysis Review Guide

Magnetic Field Foundations and Properties

• Magnetic Field Line Drawings

  • Magnetic field lines are visual representations of the magnetic force surrounding a magnet.
  • Directionality: By convention, magnetic field lines always exit from the North (N) pole and enter through the South (S) pole.
  • Internal Path: Inside the magnet, the field lines complete the loop by traveling from South back to North.
  • Density: The closer the lines are together (the higher the line density), the stronger the magnetic field in that region. The field is strongest at the poles of the magnet.

• Effects of Altering a Magnet

  • Bisection of a Bar Magnet: If a bar magnet is cut in half, the magnetic properties are preserved in both pieces. One does not obtain a separated North pole and South pole (monopoles do not exist in classical magnetism).
  • New Dipoles: Each resulting half becomes its own complete magnet with its own North and South pole. This occurs because the molecular/atomic magnetic domains remains aligned in the same direction, creating new polar interfaces at the cut surface.

• Magnetic Strength and Distance

  • Distance Relationship: The strength of a magnetic field decreases as the distance from the magnet increases.
  • Inverse-Square Law: Magnetism follows the inverse-square law, which states that the strength of the force ($B$) is inversely proportional to the square of the distance ($d$) from the source: $B \propto \frac{1}{d^2}$.
  • Halving Distance: If the distance between two magnets is halved ($d \rightarrow 0.5d$), the interaction force increases by a factor of four ($(\frac{1}{0.5})^2 = 4$).

• Permanent Magnets vs. Electromagnets

  • Permanent Magnets: These materials maintain their magnetism without an external power source. They are created by aligning the magnetic domains in a ferrous (iron-containing) material—typically by exposing it to a strong external magnetic field or by heating and cooling it within a field. Once aligned, these domains stay locked in place.
  • Electromagnets: These are temporary magnets created by running an electric current through a coil of wire (solenoid), often wrapped around a ferromagnetic core (like an iron nail). The magnetic field exists only while the current is flowing. The strength can be adjusted by changing the current or the number of loops in the coil.

• Repulsion and Magnetic Interaction

  • Same Pole Interaction: When two south poles (or two north poles) are pushed toward each other, they experience a repulsive force.
  • Field Configuration: The magnetic field lines for two repelling poles do not connect; instead, they bend away from each other, creating a region between the magnets where the net magnetic field is zero.
  • Physical Resistance: A person might not be able to get these magnets to touch because the repulsive force increases exponentially (per the inverse-square law) as they get closer, eventually exceeding the physical strength the person can apply.

Electrostatics and Electric Fields

• Electric Field Direction and Mapping

  • Single Positive Charge: Field lines radiate outward (away from the charge) in all directions.
  • Single Negative Charge: Field lines radiate inward (toward the charge) from all directions.
  • Attraction: Occurs between opposite charges (+ and -). Field lines originate on the positive charge and terminate on the negative charge.
  • Repulsion: Occurs between like charges (+ and + or - and -). Field lines bend away from each other and do not cross.

• Analyzing Field Density (Drawing E)

  • Relative Force: If location X is closer to the source charge than location Y, the field lines at X are denser. A positive test charge at location Y would experience a significantly smaller force than it would at location X.
  • Action at X: If a positive test charge is released at location X near a positive source, it will accelerate away from the source in a straight line, following the field path, because like charges repel.
  • Action at Y: If released at location Y, it will also accelerate away, but with less initial acceleration than at point X because the electric field is weaker at a greater distance.
  • Charge Polarity Change: If the source charge were negative instead of positive, the test charge (positive) would be attracted toward the source. The field lines would point inward, and the test charge would accelerate toward the source rather than away.

• Charging Methods

  • Friction: Rubbing two different materials together. Electrons are stripped from one material and transferred to the other based on the triboelectric series.
  • Conduction: Charging by direct contact. A charged object touches a neutral conductor, and the charge spreads across both until equilibrium is reached.
  • Induction: Charging without contact. A charged object is brought near (but not touching) a neutral conductor, causing a polarization of charges. If the conductor is grounded while the charged object is nearby, it will acquire a net charge opposite to the object.

• Electronegativity and the Triboelectric Effect

  • Electronegativity: A measure of the tendency of an atom or material to attract a bonding pair of electrons.
  • Balloon and Rabbit Fur Example: When a balloon is rubbed against rabbit fur, the balloon becomes negative because rubber has a higher affinity for electrons (higher electronegativity) than rabbit fur. The rabbit fur is left with a positive charge.

Coulomb's Law and Mathematical Analysis

• The Formula

  • The electrical force ($F_e$) between two point charges is calculated using: $F_e = k \frac{q_1 q_2}{d^2}$
  • Variables:
    • $k$: Coulomb's Constant ($8.99 \times 10^9\,N\,m^2/C^2$)
    • $q_1, q_2$: Quantities of the charges (in Coulombs, $C$)
    • $d$: Distance between the centers of the charges (in meters, $m$)

• Relationships

  • Force and Charge: There is a direct (linear) relationship. Doubling one charge doubles the force.
  • Force and Distance: There is an inverse-square relationship.
    • If distance is doubled ($2d$), force becomes $\frac{1}{4}$ of the original.
    • If distance is tripled ($3d$), force becomes $\frac{1}{9}$ of the original.

• Atomic Scale Comparison

  • Hydrogen Example: For a proton and electron separated by $5.0 \times 10^{-11}\,m$:
    • Charge $q = 1.6 \times 10^{-19}\,C$
    • $F_e = (8.99 \times 10^9) \frac{(1.6 \times 10^{-19}) \times (1.6 \times 10^{-19})}{(5.0 \times 10^{-11})^2} \approx 9.2 \times 10^{-8}\,N$
  • Gravity vs. Electric Force: The electrical force is vastly stronger than the force of gravity ($F_g$) between atomic particles. Gravity is only dominant at macroscopic/astronomical scales because large bodies are usually electrically neutral.

• Superposition and Net Force Calculation

  • Force Between Two Spheres: If $q_1 = +3.0 \times 10^{-6}\,C$ and $q_2 = +6.0 \times 10^{-6}\,C$ at $2\,m$:
    • $F_e = (8.99 \times 10^9) \frac{(3.0 \times 10^{-6}) \times (6.0 \times 10^{-6})}{(2)^2} = 0.040455\,N$
    • This force is repulsive because both charges are positive.
  • Net Force on Multiple Charges: To find the net force on a third charge ($q_3$) in a line, calculate the individual force from $q_1$ on $q_3$ and the force from $q_2$ on $q_3$ using their respective distances, then sum them as vectors (considering direction).

Circuit Analysis Practice

• Mixed Circuit Configuration

  • Setup: A battery ($V = 12\,V$) connected to a series resistor ($R_1 = 4\,\Omega$) which then leads to a parallel bank containing $R_2 = 6\,\Omega$, $R_3 = 3\,\Omega$, and $R_4 = 6\,\Omega$.
  • Equivalent Resistance ($R_{eq}$) Calculation:
    1. Solve the parallel part: $\frac{1}{R_p} = \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4} = \frac{1}{6} + \frac{1}{3} + \frac{1}{6} = \frac{1}{6} + \frac{2}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$. Thus, $R_p = 1.5\,\Omega$.
    2. Add the series component: $R_{total} = R_1 + R_p = 4\,\Omega + 1.5\,\Omega = 5.5\,\Omega$.
  • Total Current ($I_{total}$): Using $I = \frac{V}{R}$, $I_{total} = \frac{12\,V}{5.5\,\Omega} \approx 2.18\,A$.

• Voltage Drops and Individual Currents

  • Voltage drop across $R_1$: $V_1 = I_{total} \times R_1 = 2.18\,A \times 4\,\Omega = 8.72\,V$.
  • Voltage available for the parallel bank: $V_p = V_{total} - V_1 = 12\,V - 8.72\,V = 3.28\,V$.
  • Current through parallel resistors:
    • $I_2 = \frac{3.28\,V}{6\,\Omega} \approx 0.55\,A$
    • $I_3 = \frac{3.28\,V}{3\,\Omega} \approx 1.09\,A$
    • $I_4 = \frac{3.28\,V}{6\,\Omega} \approx 0.55\,A$

Circuit Tables and Power

• Governing Formulas

  • Ohm's Law: $V = I \times R$
  • Power Law: $P = I \times V$ or $P = I^2 \times R$

• Analytical Steps for Complex Tables

  1. Determine Total Resistance ($R_T$): Simplify the circuit into its equivalent resistance.
  2. Determine Total Current ($I_T$): Divide the source voltage by the total resistance.
  3. Distribute Voltage and Current: For series components, current is constant. For parallel components, voltage is constant.
  4. Calculate Power ($P$): Once $V$ and $I$ are known for a specific resistor, multiply them to find the power dissipation in Watts ($W$).