Lecture 5 Part A

Electric Fields

• Definition: The electric field can be understood as a disturbance in space and time created by a charge, leading to an electric disturbance that exerts forces on other charges within its vicinity.

• Analogy with Gravitational Field: Just like a gravitational field affects masses, an electric field affects charged objects without direct contact.

Force on Charges in Electric Field

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  Force Equation: The force experienced by a charge in an electric field can be described by the equation:F = qE

  • F: Force

  • q: Charge placed in the electric field

  • E: Electric field strength

• Vector Quantity: The electric field is a vector, meaning both magnitude and direction are necessary for its complete description.

Calculating Electric Field for a Point Charge

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  Electric Field Calculation: For a point charge, the electric field (E) can be derived using:E = k * |q| / r²

  • k: Coulomb's constant (9 × 10⁹ N·m²/C²)

  • |q|: Absolute value of the charge

  • r: Distance from the charge to point of interest

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  Permittivity: Coulomb's constant can also be expressed as:k = 1 / (4πε₀)

  • ε₀: Permittivity of free space (8.85 × 10⁻¹² C²/(N·m²))

• Positive Test Charge (PTC): To determine the direction of the electric field, probe with a small positive test charge.

Example: Electric Dipole

• Setup: Two point charges (q1 = +12 nC and q2 = -12 nC) are placed 0.1 meters apart.

  • Point A: Located 0.06 m from q1 and 0.04 m from q2.

  • Point B: Located 0.04 m from q1 and 0.14 m from q2.

  • Point C: Located equidistantly (0.13 m) from both charges.

Finding Electric Field at Points

1. Point A:

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     E1 due to q1:E1 = (9 × 10⁹) * (12 × 10⁻⁹) / (0.06)² = 3 × 10⁴ N/C

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     E2 due to q2:E2 = (9 × 10⁹) * (12 × 10⁻⁹) / (0.04)² = 6.8 × 10⁴ N/C

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     Total Electric Field:E_net = E1 + E2 = 9.81 × 10⁴ N/C along positive x direction.

2. Point B:

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     E1 at B (repelled by q1):E1 = -(9 × 10⁹) * (12 × 10⁻⁹) / (0.04)² = -6.8 × 10⁴ N/C

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     E2 at B (attracted by q2):E2 = (9 × 10⁹) * (12 × 10⁻⁹) / (0.14)² = 0.551 × 10⁴ N/C

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     Total Electric Field:E_net = -6.25 × 10⁴ N/C along negative x direction.

3. Point C:

   • E1 and E2 will have equal magnitudes towards each other, resulting in the y-components canceling out.

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     Net Electric Field: Only the x-components contribute:E_net = 4.92 × 10³ N/C along positive x direction.

Electric Field Lines

• Purpose: Visual representation of the electric field strength and direction.

Rules for Drawing Electric Field Lines

1. Direction: Lines begin at positive charges and end at negative charges.

2. Density: The density of lines indicates the electric field's strength; more lines = stronger field.

3. Tangents: The direction of electric field vectors at any point is tangent to the field line.

4. No Crossing: Electric field lines never cross, indicating that each point has a unique electric field direction.

Conclusion

• To analyze electric fields: Probe with a positive test charge, calculate components, sum vectorially, and utilize electric field lines for a visual representation of the electric field behavior.