phy2

Electric Circuitry

I. Definition of Terms

• Equipotential Surface

  • Definition: A region in space where points in the electric field have equal potential.
  • Representation: Visualized by lines surrounding a point charge.

• Electric Potential Energy

  • Definition: The potential energy associated with a moving charge changing position within an electric field produced by a stationary charge.

• Electric Potential

  • Also known as: Voltage.
  • Definition: Electric potential energy per unit charge, which arises due to the placement of a charge within an electric field.
  • Unit: Volt (𝑉).
  • Relationship: 1 𝑉 = 1 𝐽/𝐶.

• Potential Difference

  • Definition: The difference between two electric potential values.
  • Importance: Responsible for the generation of electric current.
  • Note: Earth's potential is arbitrarily defined as zero (0).

• Work (Electric)

  • Definition: Measure of energy transfer between electric charges; refers to the influence of a stationary charge upon a moving charge in its electric field.
  • Characteristics:
  • Influence of electric forces within a specific radius.
  • Cases of Work (𝑊) and Electric Potential Energy (𝑈𝐸):
    • If 𝑞0 in 𝐸 & 𝑞 = +, then 𝑊 > 0 and 𝑈𝐸 < 0.
    • If 𝑞0 in 𝐸 = +, 𝑞 = -, then 𝑊 < 0 and 𝑈𝐸 > 0.
    • If 𝑞0 in 𝐸 & 𝑞 = -, then 𝑊 < 0 and 𝑈𝐸 > 0.
    • If 𝑞0 = - and 𝑞 = +, then 𝑊 > 0 and 𝑈𝐸 < 0.

• Potential Gradient

  • Definition: The local rate of change of potential with respect to displacement.
  • Behavior:
  • Positive charges are accelerated down gradients.
  • Negative charges are accelerated up gradients.

II. Capacitor Concepts

• Capacitor

  • Also known as: A condenser.
  • Definition: An electrical component that stores electrical energy in an electric field located between its parallel plates.
  • Configuration: Plates may be separated by a vacuum or an insulator.

• Capacitance

  • Definition: The ability of a capacitor to store charge, which alters the voltage of a body by one unit.
  • Unit: Farad (𝐹).
  • Relationship: 1 𝐹 = 1 𝐶/𝑉.
  • Factors affecting capacitance:
  • Area of the plates.
  • Distance separating the plates.
  • Nature of the insulating materials.
  • Uniformity of the electric field within the plates’ space.

Energy on Capacitors

1. Stored Energy
   • Definition: The energy stored due to half of the work performed.
2. Electric Field Energy
   • Definition: The energy of the electric field, equivalent to the electric field's energy density.

• Dielectric

  • Definition: An insulating material inserted between a capacitor's parallel plates that increases capacitance without affecting the separation distance of the plates.

• Equivalent Capacitance

  • Definition: The total capacitance within sets of given capacitors representing the overall capacitance of all capacitors within a circuit.
  • Series Circuit: Connected devices share charge values in a sequential manner.
  • Parallel Circuit: Connected devices operate independently from one another, sharing voltage values.

III. FORMULAE

• Electric Potential Energy

  • Formula:
    $U<em>E = k \frac{qq</em>0}{r}$
  • Notes:
    • Where:
    • 𝑟 = radius between charges.
    • 𝑞 = moving charge.
    • 𝑞0 = rest charge.

• Electric Potential

  • Formula:
    $U<em>E = q</em>0 V$
  • Electric Potential (with permittivity):
    $V = k \frac{q}{r}$
  • Electric Potential (if electric field is given):
    $V = E imes r$
  • Electric Potential (if force and charge are given):
    $V = \frac{F_E}{q} imes r$

• Potential Difference

  • Formula:
    $V<em>d = riangle V$$V</em>d = V - V_0$

• Electric Work

  • Formulas:
    $W<em>E = q riangle V$$W</em>E = F_E imes r$

• Gradient

  • Formula:
    $F = \frac{ riangle ext{ϕ}}{ riangle r}$
  • Where:
    • ϕ = scalar potential.

• Field Vector

  • Formula:
    
    abla = ar{i} + ar{j} + ar{k}
    
  • Where:
    • $ar{i}$ = unit vector (x-axis).
    • $ar{j}$ = unit vector (y-axis).
    • $ar{k}$ = unit vector (z-axis).

• Electric Field as a Potential Gradient

  • Formula:
    $E = -<br />\nabla V$
    $E = -\bigg[\bar{i} \frac{ ext{∂}}{ ext{∂x}} + \bar{j} \frac{ ext{∂}}{ ext{∂y}} + \bar{k} \frac{ ext{∂}}{ ext{∂z}}\bigg] V$

• Capacitance

  • Formula:
    $C = \frac{q}{V_d}$
  • Capacitance (given area and material permittivity):
    $C = \frac{ ext{ε} A}{r}$
  • Where
    • 𝜀 = material permittivity.
  • Capacitance (given charge density):
    $C = \frac{qr}{E}$
  • Where:
    • $E = \frac{ ext{σ}}{ ext{ε}}$
    • $ext{σ} = charge density = \frac{q}{A}$

• Capacitance (given the dielectric)

  • Formula:
    $C = k ext{ε}_0 \frac{A}{r}$
  • Where:
    • 𝜀₀ = vacuum permittivity = 8.85 × 10^{-12} 𝐹/m.
    • 𝑘 = relative dielectric permittivity.

Stored Energy

• Stored Energy
  • Formula:
    $U_C = \frac{q^2}{2C}$
  • Stored Energy (given Voltage):
    $U_C = \frac{q V}{2}$
  • Stored Energy (given Capacitance and Voltage):
    $U_C = \frac{C V^2}{2}$

Electric Field Energy

• Electric Field Energy
  • Formula:
    $T<em>E = ext{η}</em>C V_o$
  • Where:
    • 𝜂₵ = energy density.
    • 𝑉ₒ = volume.

Equivalent Capacitance

• Series Capacitance
  • Conditions:
    $q<em>T = q</em>1 = q<em>2 = … = q</em>n$
    $V<em>T = V</em>1 + V<em>2 + … + V</em>n$
    $\frac{1}{C<em>T} = \frac{1}{C</em>1} + \frac{1}{C<em>2} + … + \frac{1}{C</em>n}$
• Parallel Capacitance
  • Conditions:
    $q<em>T = q</em>1 + q<em>2 + … + q</em>n$
    $V<em>T = V</em>1 = V<em>2 = … = V</em>n$
    $C<em>T = C</em>1 + C<em>2 + … + C</em>n$

References

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