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.
**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.
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: Electric field units are newtons per coulomb (N/C).
This ratio derives from the force (N) divided by the test charge (C).
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.
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.
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):
Force experienced by the electron: [ F = qE ]
Acceleration: [ a = \frac{F}{m} ] = [ \frac{qE}{m} ]
Substituting values yields acceleration.
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.
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.
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.