Math Problem Statement

Please solve

An electric dipole is placed on the y-axis with +Q at y = +d and -Q at y = -d. Calculate the magnitude of the electric field at point P located at a distance d on the x-axis.

Solution

To calculate the electric field at point PP due to an electric dipole on the yy-axis, let's analyze the setup and apply the relevant equations.

Problem Setup

  • There is a dipole on the yy-axis with charges:
    • +Q+Q at y=+dy = +d
    • Q-Q at y=dy = -d
  • The point PP where we want to calculate the electric field is on the xx-axis, a distance dd from the origin.

Solution

  1. Determine Distance to Point P:

    • The distance from each charge ±Q\pm Q to point PP is r=d2+d2=2dr = \sqrt{d^2 + d^2} = \sqrt{2}d.
  2. Calculate Electric Field Due to Each Charge:

    • The electric field EE due to a point charge QQ at a distance rr is: E=kQr2=kQ(2d2)E = \frac{k |Q|}{r^2} = \frac{k |Q|}{(2d^2)} where k=14πϵ0k = \frac{1}{4 \pi \epsilon_0} is Coulomb's constant.
  3. Component Analysis:

    • Since +Q+Q and Q-Q are symmetrically placed around the origin, their yy-components of the electric field at PP will cancel out. We only need the xx-components.
    • The xx-component of each field ExE_x is given by: Ex=Ecosθ=kQ2d2d2d=kQ2d212=kQ22d2E_x = E \cos \theta = \frac{k |Q|}{2d^2} \cdot \frac{d}{\sqrt{2}d} = \frac{k |Q|}{2d^2} \cdot \frac{1}{\sqrt{2}} = \frac{k |Q|}{2\sqrt{2} d^2}
  4. Total Electric Field at Point P:

    • Since both +Q+Q and Q-Q contribute equally in the xx-direction, the total electric field EtotalE_{\text{total}} at PP is: Etotal=2Ex=2kQ22d2=kQ2d2E_{\text{total}} = 2 \cdot E_x = 2 \cdot \frac{k |Q|}{2\sqrt{2} d^2} = \frac{k |Q|}{\sqrt{2} d^2}

Final Answer

The magnitude of the electric field at point PP is: Etotal=kQ2d2E_{\text{total}} = \frac{k |Q|}{\sqrt{2} d^2}

Would you like further explanation on any part of this solution?

Follow-up Questions:

  1. What would the electric field be if PP were twice the distance on the xx-axis?
  2. How does the direction of the electric field change if PP is located at x=dx = -d?
  3. What would happen to the electric field if the dipole charges were doubled?
  4. How does the electric field vary along the yy-axis?
  5. How does this solution change if we consider the electric potential at point PP instead of the electric field?

Tip:

For electric dipoles, always consider symmetry to simplify component calculations, especially when fields in one direction cancel out.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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