Question

Two point charges -Q and Q are located at points (d, 0) and (0, d) respectively, in x-y plane. The electric field E at the origin will be:

• A. \(\frac{1}{4\pi\epsilon_0} \frac{\sqrt{2}Q}{d^2} (\hat{i} - \hat{j})\)
• B. \(\frac{1}{4\pi\epsilon_0} \frac{\sqrt{2}Q}{d^2} (-\hat{i} - \hat{j})\)
• C. \(\frac{1}{4\pi\epsilon_0} \frac{Q}{d^2} (\hat{i} - \hat{j})\)
• D. \(\frac{1}{4\pi\epsilon_0} \frac{Q}{d^2} (-\hat{i} - \hat{j})\)

Hint:

Always draw the vectors first! For the origin, a field from a positive charge on the y-axis points down (\(-\hat{j}\)), and a field from a negative charge on the x-axis points right (\(+\hat{i}\)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The field at the origin is the vector sum of the fields of the two charges.
A field points away from a positive charge and towards a negative charge.

Step 2: Key Formula or Approach:
For a point charge, \( E = \frac{1}{4\pi\epsilon_0} \frac{|q|}{r^2} \). Both charges are at distance \( d \) from the origin.

Step 3: Detailed Explanation:
1. The charge \( -Q \) is at \( (d, 0) \). It is negative, so its field at the origin points towards the charge, along \( +\hat{i} \): \( \vec{E}_1 = \frac{1}{4\pi\epsilon_0} \frac{Q}{d^2} \hat{i} \).
2. The charge \( +Q \) is at \( (0, d) \). It is positive, so its field at the origin points away from the charge, along \( -\hat{j} \): \( \vec{E}_2 = -\frac{1}{4\pi\epsilon_0} \frac{Q}{d^2} \hat{j} \).
3. Adding the two fields: \[ \vec{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{d^2} (\hat{i} - \hat{j}) \]
The magnitude is \( \sqrt{2} \frac{Q}{4\pi\epsilon_0 d^2} \), because the two components are equal and perpendicular. The factor \( \sqrt{2} \) is already contained in the vector \( (\hat{i} - \hat{j}) \), whose length is \( \sqrt{2} \).

Step 4: Checking the options:
Option (A) has an extra factor \( \sqrt{2} \), so its magnitude is \( 2\frac{Q}{4\pi\epsilon_0 d^2} \). This is too large.
Option (B) has the extra \( \sqrt{2} \) and the wrong direction \( (-\hat{i} - \hat{j}) \).
Option (D) has the wrong direction \( (-\hat{i} - \hat{j}) \).
Option (C) matches the result exactly.

Step 5: Final Answer:
\( \vec{E} = \frac{1}{4\pi\epsilon_0} \frac{Q}{d^2} (\hat{i} - \hat{j}) \), which is option (C).