What is the magnitude of the net electric field at the center of a square due to four charges in terms of q? Express your answer in terms of variables.
Understand the Problem
The question is asking for the magnitude of the net electric field at the center of a square due to four charges. It requests the answer to be expressed in terms of the variable 'q'. This implies a physics problem related to electric fields and charge configurations.
Answer
The magnitude of the net electric field at the center of the square is $E_{\text{net}} = \frac{4k |q|}{a^2}$.
Answer for screen readers
The magnitude of the net electric field at the center of the square due to the four charges is:
$$ E_{\text{net}} = \frac{2\sqrt{2} k |q|}{\left( \frac{a}{\sqrt{2}} \right)^2} = \frac{4k |q|}{a^2} $$
Steps to Solve
- Identify the configuration of charges
Each corner of the square has a charge $q$. Since the square has four corners, we have four charges positioned at these corners.
- Calculate electric field due to one charge
The electric field due to a single point charge $q$ at a distance $r$ is given by the formula:
$$ E = \frac{k |q|}{r^2} $$
where $k$ is Coulomb's constant, approximately $8.99 \times 10^9 , \text{Nm}^2/\text{C}^2$.
- Determine the distance from a charge to the center of the square
For a square of side length $a$, the distance $r$ from any charge to the center can be calculated using the Pythagorean theorem:
$$ r = \frac{a}{\sqrt{2}} $$
- Calculate electric field due to all charges
The total electric field at the center is the vector sum of the fields created by each charge. Since the charges are equal and symmetrically placed, the horizontal and vertical components of the fields can be calculated.
- Find components of the electric field
From one charge, the electric field's components in the x and y directions are:
$$ E_{x} = E \cdot \cos(45°) = \frac{k |q|}{r^2} \cdot \frac{1}{\sqrt{2}} $$
$$ E_{y} = E \cdot \sin(45°) = \frac{k |q|}{r^2} \cdot \frac{1}{\sqrt{2}} $$
- Sum the x and y components of the fields
Since there are two charges contributing to each component, we double the contributions:
$$ E_{x, \text{total}} = 2 \cdot E_{x} = 2 \cdot \frac{k |q|}{r^2} \cdot \frac{1}{\sqrt{2}} $$
$$ E_{y, \text{total}} = 2 \cdot E_{y} = 2 \cdot \frac{k |q|}{r^2} \cdot \frac{1}{\sqrt{2}} $$
Since the electric fields are identical, the overall electric field can be expressed as:
$$ E_{\text{net}} = E_{x, \text{total}} + E_{y, \text{total}} = \sqrt{E_{x, \text{total}}^2 + E_{y, \text{total}}^2} $$
- Substituting values
Substituting the expressions for $E_{x, \text{total}}$ and $E_{y, \text{total}}$ and simplifying gives us the net electric field at the center of the square.
The magnitude of the net electric field at the center of the square due to the four charges is:
$$ E_{\text{net}} = \frac{2\sqrt{2} k |q|}{\left( \frac{a}{\sqrt{2}} \right)^2} = \frac{4k |q|}{a^2} $$
More Information
The net electric field produced by point charges can vary significantly depending on their arrangement. Here, the symmetry of the square allows for a simplified calculation. Notably, the magnitude of the electric field experienced by a charge depends upon both the value of the charge and the distance from the source of the electric field.
Tips
- Not considering the direction of the electric fields. Since the charges are positive, the fields point away from the charges.
- Incorrectly measuring the distance $r$ to the center of the square.
- Neglecting to simplify compound expressions properly leading to incorrect final answers.
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