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To derive an expression for the magnitude \( E \) of the electric field of the dipole at the evaluation point, we consider the following:

The dipole consists of charges \( +q \) and \( -q \), separated by a distance \( d \). The evaluation point is a distance \( r \) from the dipole center in the y-direction, and we are assuming that \( r \) is much larger than \( d \) (i.e., \( r \gg d \)).

1. **Electric Field of a Dipole**:
   The electric field \( \vec{E} \) at a point due to a dipole can be approximated, at distances much greater than the separation of the charges (which is our case), by the expression:
   \[
   \vec{E} = \frac{1}{4\pi \epsilon_0} \frac{p}{r^3} (2 \cos \theta \hat{r} + \sin \theta \hat{\theta})
   \]
   where \( p = qd \) is the dipole moment, \( \theta \) is the angle between the dipole axis and the line connecting the dipole to the point of interest, and \( \hat{r} \) and \( \hat{\theta} \) are unit vectors in the radial and angular directions, respectively.

2. **Simplifying for Our Configuration**:
   In this scenario, the dipole is oriented along the x-axis, and the point of interest is along the y-axis directly above the center of the dipole. Therefore, \( \theta = 90^\circ \) and \( \cos 90^\circ = 0 \), \( \sin 90^\circ = 1 \). The electric field simplifies to:
   \[
   \vec{E} = \frac{1}{4\pi \epsilon_0} \frac{p}{r^3} \sin \theta \hat{\theta}
   \]
   Replacing \( \theta \) by \( 90^\circ \) gives:
   \[
   \vec{E} = \frac{1}{4\pi \epsilon_0} \frac{qd}{r^3} \hat{\theta}
   \]
   
3. **Magnitude of the Electric Field**:
   The magnitude \( E \) of the electric field is:
   \[
   E = \frac{1}{4\pi \epsilon_0} \frac{qd}{r^3}
   \]

This is the expression for the magnitude of the electric field of the dipole at the evaluation point in terms of \( q \), \( d \), \( r \), and \( \epsilon_0 \).

Would you like to see how we could use this expression with actual values or have any specific questions about the steps? Here are some related questions:

1. How would the electric field change if the separation \( d \) between the charges were doubled?
2. What happens to the electric field if the charge \( q \) is halved?
3. How does the direction of the electric field change if the evaluation point is moved to different positions relative to the dipole?
4. Could you explain how the concept of the dipole moment \( p \) influences the electric field?
5. Why does the approximation \( r \gg d \) simplify the calculation of the electric field?

**Tip:** When dealing with dipoles, always consider the symmetry of the problem to simplify your calculations.

Derive an expression for the magnitude E of the electric field of the dipole at the evaluation point. The dipole consists of charges +q and -q separated by a distance d along the x-axis, and the evaluation point is a distance r away along the y-axis, where r >> d.

Learn how to derive the electric field of a dipole consisting of charges +q and -q separated by distance d along the x-axis, with an evaluation point along the y-axis. This problem uses electrostatic concepts and electric field equations, including dipole moment and field approximations for r >> d.

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