What is the relationship between electric field and the distance between the point charge and a specific point in space?
2 Answers
The relationship between the electric field \( \mathbf{E} \) and the distance \( r \) from a point charge \( Q \) is described by Coulomb's law. To understand this relationship clearly, let's break it down step by step:
### 1. **Electric Field Definition**
The electric field \( \mathbf{E} \) at a point in space due to a point charge \( Q \) is defined as the force \( \mathbf{F} \) per unit positive charge \( q \) placed at that point. Mathematically, this is given by:
\[ \mathbf{E} = \frac{\mathbf{F}}{q} \]
where \( \mathbf{F} \) is the force experienced by the charge \( q \) due to \( Q \).
### 2. **Coulomb’s Law**
According to Coulomb’s law, the magnitude of the electric field \( E \) created by a point charge \( Q \) at a distance \( r \) from the charge is:
\[ E = \frac{k |Q|}{r^2} \]
where:
- \( E \) is the magnitude of the electric field.
- \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \).
- \( |Q| \) is the absolute value of the charge \( Q \).
- \( r \) is the distance from the point charge to the point where the field is being measured.
### 3. **Inverse Square Law**
The formula \( E = \frac{k |Q|}{r^2} \) reveals that the electric field \( E \) varies inversely with the square of the distance \( r \). This is known as the inverse square law. Specifically:
- **As the distance \( r \) increases**, the electric field \( E \) decreases. The relationship is quadratic, so if you double the distance, the electric field becomes one-fourth of its original value.
- **As the distance \( r \) decreases**, the electric field \( E \) increases. For example, halving the distance will quadruple the electric field.
### 4. **Direction of the Electric Field**
The electric field is a vector quantity and has both magnitude and direction. The direction of the electric field \( \mathbf{E} \) due to a positive point charge \( Q \) is radially outward from the charge. Conversely, for a negative charge \( Q \), the electric field points radially inward toward the charge.
### 5. **Field Lines and Intensity**
In terms of field lines, the density of electric field lines in a region is proportional to the magnitude of the electric field in that region. Closer to the charge, field lines are more densely packed, indicating a stronger electric field. As you move further away from the charge, the lines spread out, and the electric field becomes weaker.
### Summary
To summarize, the electric field \( E \) due to a point charge \( Q \) is inversely proportional to the square of the distance \( r \) from the charge. This relationship is fundamental to electrostatics and explains how the influence of a charge diminishes with distance.
### 1. **Electric Field Definition**
The electric field \( \mathbf{E} \) at a point in space due to a point charge \( Q \) is defined as the force \( \mathbf{F} \) per unit positive charge \( q \) placed at that point. Mathematically, this is given by:
\[ \mathbf{E} = \frac{\mathbf{F}}{q} \]
where \( \mathbf{F} \) is the force experienced by the charge \( q \) due to \( Q \).
### 2. **Coulomb’s Law**
According to Coulomb’s law, the magnitude of the electric field \( E \) created by a point charge \( Q \) at a distance \( r \) from the charge is:
\[ E = \frac{k |Q|}{r^2} \]
where:
- \( E \) is the magnitude of the electric field.
- \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2 \).
- \( |Q| \) is the absolute value of the charge \( Q \).
- \( r \) is the distance from the point charge to the point where the field is being measured.
### 3. **Inverse Square Law**
The formula \( E = \frac{k |Q|}{r^2} \) reveals that the electric field \( E \) varies inversely with the square of the distance \( r \). This is known as the inverse square law. Specifically:
- **As the distance \( r \) increases**, the electric field \( E \) decreases. The relationship is quadratic, so if you double the distance, the electric field becomes one-fourth of its original value.
- **As the distance \( r \) decreases**, the electric field \( E \) increases. For example, halving the distance will quadruple the electric field.
### 4. **Direction of the Electric Field**
The electric field is a vector quantity and has both magnitude and direction. The direction of the electric field \( \mathbf{E} \) due to a positive point charge \( Q \) is radially outward from the charge. Conversely, for a negative charge \( Q \), the electric field points radially inward toward the charge.
### 5. **Field Lines and Intensity**
In terms of field lines, the density of electric field lines in a region is proportional to the magnitude of the electric field in that region. Closer to the charge, field lines are more densely packed, indicating a stronger electric field. As you move further away from the charge, the lines spread out, and the electric field becomes weaker.
### Summary
To summarize, the electric field \( E \) due to a point charge \( Q \) is inversely proportional to the square of the distance \( r \) from the charge. This relationship is fundamental to electrostatics and explains how the influence of a charge diminishes with distance.
The relationship between the electric field (\(E\)) and the distance (\(r\)) between a point charge (\(q\)) and a specific point in space can be described using **Coulomb's law**.
For a **point charge**, the magnitude of the electric field at a distance \(r\) from the charge is given by:
\[
E = \frac{k_e \cdot |q|}{r^2}
\]
Where:
- \(E\) is the electric field strength.
- \(k_e\) is Coulomb's constant (\(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\)).
- \(q\) is the magnitude of the point charge.
- \(r\) is the distance from the point charge to the point in space where the electric field is being measured.
### Key Insights:
1. **Inverse Square Law**: The electric field strength is **inversely proportional** to the square of the distance (\(r^2\)) between the point charge and the point in space. This means that as the distance increases, the electric field strength decreases rapidly.
2. **Proportional to Charge**: The electric field is **directly proportional** to the magnitude of the charge. A larger charge generates a stronger electric field.
### Physical Interpretation:
- **Closer to the charge**: As \(r\) decreases (i.e., moving closer to the charge), the electric field increases significantly.
- **Farther from the charge**: As \(r\) increases (i.e., moving farther away), the electric field decreases rapidly due to the \(1/r^2\) dependence.
In summary, the electric field due to a point charge becomes weaker with the square of the distance from the charge, highlighting the importance of proximity in the strength of electric fields.
For a **point charge**, the magnitude of the electric field at a distance \(r\) from the charge is given by:
\[
E = \frac{k_e \cdot |q|}{r^2}
\]
Where:
- \(E\) is the electric field strength.
- \(k_e\) is Coulomb's constant (\(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\)).
- \(q\) is the magnitude of the point charge.
- \(r\) is the distance from the point charge to the point in space where the electric field is being measured.
### Key Insights:
1. **Inverse Square Law**: The electric field strength is **inversely proportional** to the square of the distance (\(r^2\)) between the point charge and the point in space. This means that as the distance increases, the electric field strength decreases rapidly.
2. **Proportional to Charge**: The electric field is **directly proportional** to the magnitude of the charge. A larger charge generates a stronger electric field.
### Physical Interpretation:
- **Closer to the charge**: As \(r\) decreases (i.e., moving closer to the charge), the electric field increases significantly.
- **Farther from the charge**: As \(r\) increases (i.e., moving farther away), the electric field decreases rapidly due to the \(1/r^2\) dependence.
In summary, the electric field due to a point charge becomes weaker with the square of the distance from the charge, highlighting the importance of proximity in the strength of electric fields.