Math Problem Statement
Please solve
Solution
To calculate the electric field at point due to an electric dipole on the -axis, let's analyze the setup and apply the relevant equations.
Problem Setup
- There is a dipole on the -axis with charges:
- at
- at
- The point where we want to calculate the electric field is on the -axis, a distance from the origin.
Solution
-
Determine Distance to Point P:
- The distance from each charge to point is .
-
Calculate Electric Field Due to Each Charge:
- The electric field due to a point charge at a distance is: where is Coulomb's constant.
-
Component Analysis:
- Since and are symmetrically placed around the origin, their -components of the electric field at will cancel out. We only need the -components.
- The -component of each field is given by:
-
Total Electric Field at Point P:
- Since both and contribute equally in the -direction, the total electric field at is:
Final Answer
The magnitude of the electric field at point is:
Would you like further explanation on any part of this solution?
Follow-up Questions:
- What would the electric field be if were twice the distance on the -axis?
- How does the direction of the electric field change if is located at ?
- What would happen to the electric field if the dipole charges were doubled?
- How does the electric field vary along the -axis?
- How does this solution change if we consider the electric potential at point instead of the electric field?
Tip:
For electric dipoles, always consider symmetry to simplify component calculations, especially when fields in one direction cancel out.
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Math Problem Analysis
Mathematical Concepts
Electric Fields
Dipoles
Vector Components
Formulas
Electric field due to a point charge: E = kQ / r^2
Distance from each charge to point P: r = √(2)d
Total electric field from dipole at point P: E_total = kQ / (√2 * d^2)
Theorems
Superposition Principle for Electric Fields
Suitable Grade Level
College-level Physics