physics 02-14

Introduction to Vectors and Vector Fields

• Definition of a Vector: A quantity having both magnitude and direction.

• Vector Fields: A function that assigns a vector to every point in space. Every point in a vector field can be represented by an arrow indicating the direction and magnitude of a vector.

  • Example: The electric field is a vector field.

Understanding Electric Fields

• **Electric Field Concept: **

  • Defined at a point in space based on the force felt by a test charge at that point.

  • Place a charge at a specific point; measure the force (F) it experiences, then divide that by the magnitude of the test charge (q) used.

  • Formula: [ E = \frac{F}{q_{test}} ]

  • This defines the electric field at point P.

Calculating Electric Field

• Force Measurement:

  • Use any test charge (e.g., 10 microcoulombs). Measure the force it feels in the electric field.

  • The calculated ratio of force to charge is unique to that point in the electric field.

• Key Point: Regardless of the size or sign of the test charge, the electric field measured remains constant at that point.

Units of Electric Field

• Units: Electric field units are newtons per coulomb (N/C).

  • This ratio derives from the force (N) divided by the test charge (C).

Direction of Forces on Charges in Electric Fields

• Behavior of Positive Charges:

  • When a positive charge enters an electric field, it experiences a force in the direction of the electric field (parallel).

  • Hence, a positive charge accelerates in the direction of the electric field.

• Behavior of Negative Charges:

  • A negative charge experiences force opposite to the direction of the electric field (antiparallel).

  • Therefore, it accelerates against the direction of the electric field.

Example of Electric Fields in Nature

• Earth's Electric Field:

  • At the core of the Earth, hot plasma creates a dynamic, charged environment.

  • The movement of these charged particles generates electric and magnetic fields around the Earth.

  • Situational Scenario: Thunderclouds also create local electric fields at Earth's surface.

Quantitative Analysis

• Sample Calculation: Given the electric field value (e.g., 300 N/C) and its direction, find acceleration of an electron (q = -1.6 × 10^-19 C, m = 9.1 × 10^-31 kg):

  1. Force experienced by the electron: [ F = qE ]

  2. Acceleration: [ a = \frac{F}{m} ] = [ \frac{qE}{m} ]

  3. Substituting values yields acceleration.

Electric Field from Point Charges

• Point Charge and Electric Field:

  • The electric field produced by a point charge (q) at a distance r is given by: [ E = k \frac{|q|}{r^2} ]

  • Where k is Coulomb's constant.

  • Direction: For a positive charge, the electric field points radially outward; for a negative charge, it points radially inward.

Superposition Principle for Multiple Charges

• Electric Field from Multiple Point Charges:

  • The net electric field at a point due to multiple charges is the vector sum of the individual fields produced by each charge.

  • Use the superposition principle to calculate total electric field given various charge configurations:

    • Example: If there are two different charges at specific coordinates, calculate the electric field at a point by summing effects from both charges.

Conclusion

• Understanding electric fields is crucial for analyzing electric forces and interactions between charged particles.

• The ratio of force to charge gives a concise method of defining the electric field at any point in space.