What is the relation between electric field at a point and its distance from short dipole?
2 Answers
The relation between the electric field at a point and its distance from a **short dipole** (also called an ideal dipole) is determined by the nature of the dipole's electric field distribution.
### Understanding a Short Dipole
A short dipole consists of two equal and opposite charges \( +q \) and \( -q \), separated by a small distance \( d \). The dipole moment \( p \) is given by:
\[
p = q \cdot d
\]
where:
- \( p \) is the dipole moment (a vector quantity),
- \( q \) is the charge magnitude,
- \( d \) is the separation distance between the charges.
### Electric Field of a Dipole
The electric field due to a short dipole depends on the point of observation and its distance from the dipole. There are two primary regions to consider:
1. **Axial Line (along the axis of the dipole)**: The point lies on the extended line passing through the dipole (along the dipole moment).
2. **Equatorial Line (perpendicular to the dipole axis)**: The point lies on a line that is perpendicular to the dipole axis and passes through the center of the dipole.
#### 1. Electric Field on the Axial Line
At a point located along the dipole axis (at a distance \( r \) from the center of the dipole), the electric field \( E_{\text{axial}} \) is given by:
\[
E_{\text{axial}} = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{2p}{r^3}
\]
where:
- \( \varepsilon_0 \) is the permittivity of free space,
- \( p \) is the dipole moment,
- \( r \) is the distance from the dipole.
This shows that the electric field on the axial line **varies inversely with the cube of the distance** from the dipole, i.e., \( E \propto \frac{1}{r^3} \).
#### 2. Electric Field on the Equatorial Line
At a point on the equatorial line (at a distance \( r \) from the center of the dipole), the electric field \( E_{\text{equatorial}} \) is given by:
\[
E_{\text{equatorial}} = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{p}{r^3}
\]
Here again, the electric field varies **inversely with the cube of the distance** from the dipole, i.e., \( E \propto \frac{1}{r^3} \), but it is half the magnitude compared to the field along the axial line, and it points in the opposite direction of the dipole moment.
### Summary of the Relation
- On both the **axial** and **equatorial** lines, the electric field due to a short dipole follows an inverse cubic relationship with distance from the dipole:
\[
E \propto \frac{1}{r^3}
\]
- On the axial line, the field is stronger and aligned along the dipole moment, whereas on the equatorial line, the field is weaker and directed opposite to the dipole moment.
This inverse cubic relationship is a distinguishing feature of the electric field from a dipole, as opposed to a single point charge, where the electric field follows an inverse square law.
### Understanding a Short Dipole
A short dipole consists of two equal and opposite charges \( +q \) and \( -q \), separated by a small distance \( d \). The dipole moment \( p \) is given by:
\[
p = q \cdot d
\]
where:
- \( p \) is the dipole moment (a vector quantity),
- \( q \) is the charge magnitude,
- \( d \) is the separation distance between the charges.
### Electric Field of a Dipole
The electric field due to a short dipole depends on the point of observation and its distance from the dipole. There are two primary regions to consider:
1. **Axial Line (along the axis of the dipole)**: The point lies on the extended line passing through the dipole (along the dipole moment).
2. **Equatorial Line (perpendicular to the dipole axis)**: The point lies on a line that is perpendicular to the dipole axis and passes through the center of the dipole.
#### 1. Electric Field on the Axial Line
At a point located along the dipole axis (at a distance \( r \) from the center of the dipole), the electric field \( E_{\text{axial}} \) is given by:
\[
E_{\text{axial}} = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{2p}{r^3}
\]
where:
- \( \varepsilon_0 \) is the permittivity of free space,
- \( p \) is the dipole moment,
- \( r \) is the distance from the dipole.
This shows that the electric field on the axial line **varies inversely with the cube of the distance** from the dipole, i.e., \( E \propto \frac{1}{r^3} \).
#### 2. Electric Field on the Equatorial Line
At a point on the equatorial line (at a distance \( r \) from the center of the dipole), the electric field \( E_{\text{equatorial}} \) is given by:
\[
E_{\text{equatorial}} = \frac{1}{4 \pi \varepsilon_0} \cdot \frac{p}{r^3}
\]
Here again, the electric field varies **inversely with the cube of the distance** from the dipole, i.e., \( E \propto \frac{1}{r^3} \), but it is half the magnitude compared to the field along the axial line, and it points in the opposite direction of the dipole moment.
### Summary of the Relation
- On both the **axial** and **equatorial** lines, the electric field due to a short dipole follows an inverse cubic relationship with distance from the dipole:
\[
E \propto \frac{1}{r^3}
\]
- On the axial line, the field is stronger and aligned along the dipole moment, whereas on the equatorial line, the field is weaker and directed opposite to the dipole moment.
This inverse cubic relationship is a distinguishing feature of the electric field from a dipole, as opposed to a single point charge, where the electric field follows an inverse square law.
The electric field \(\mathbf{E}\) due to a short dipole at a point in space depends on the distance from the dipole and the orientation of the point relative to the dipole. Here’s a detailed explanation of the relationship between the electric field and the distance from a short dipole:
### Dipole Definition
A short dipole consists of two opposite charges, \(+q\) and \(-q\), separated by a small distance \(d\). The dipole moment \(\mathbf{p}\) is defined as:
\[ \mathbf{p} = q \cdot \mathbf{d} \]
where \(\mathbf{d}\) is the vector pointing from the negative charge to the positive charge.
### Electric Field of a Dipole
The electric field \(\mathbf{E}\) of a dipole at a point in space can be analyzed in two regions: along the axial line (the line extending from the dipole along the direction of \(\mathbf{p}\)) and along the equatorial line (perpendicular to the direction of \(\mathbf{p}\)).
#### 1. **Axial Line (On-axis)**
For a point on the axial line at a distance \(r\) from the center of the dipole, the electric field is given by:
\[ E_{\text{axial}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{3p}{r^3} \]
where \(\epsilon_0\) is the permittivity of free space.
#### 2. **Equatorial Line (Off-axis)**
For a point on the equatorial line, which is perpendicular to the axis of the dipole and at a distance \(r\) from the center of the dipole, the electric field is:
\[ E_{\text{equatorial}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{r^3} \]
### Key Relationships
1. **Distance Dependence**: In both cases, the electric field decreases with the cube of the distance from the dipole. This is a characteristic feature of the dipole field, as opposed to the \(1/r^2\) dependence seen in the field of a single point charge.
2. **Dipole Moment Influence**: The electric field also depends on the magnitude of the dipole moment \(p\). A larger dipole moment results in a stronger field at a given distance.
3. **Orientation Dependence**: The field strength also depends on the position relative to the dipole's orientation. The field is strongest along the axial line and weaker on the equatorial line.
### Summary
- **On the axial line**: The electric field \(E_{\text{axial}}\) is proportional to \( \frac{3p}{r^3} \).
- **On the equatorial line**: The electric field \(E_{\text{equatorial}}\) is proportional to \( \frac{p}{r^3} \).
In essence, the electric field of a short dipole decreases with the cube of the distance from the dipole, and its strength varies depending on the direction relative to the dipole's orientation.
### Dipole Definition
A short dipole consists of two opposite charges, \(+q\) and \(-q\), separated by a small distance \(d\). The dipole moment \(\mathbf{p}\) is defined as:
\[ \mathbf{p} = q \cdot \mathbf{d} \]
where \(\mathbf{d}\) is the vector pointing from the negative charge to the positive charge.
### Electric Field of a Dipole
The electric field \(\mathbf{E}\) of a dipole at a point in space can be analyzed in two regions: along the axial line (the line extending from the dipole along the direction of \(\mathbf{p}\)) and along the equatorial line (perpendicular to the direction of \(\mathbf{p}\)).
#### 1. **Axial Line (On-axis)**
For a point on the axial line at a distance \(r\) from the center of the dipole, the electric field is given by:
\[ E_{\text{axial}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{3p}{r^3} \]
where \(\epsilon_0\) is the permittivity of free space.
#### 2. **Equatorial Line (Off-axis)**
For a point on the equatorial line, which is perpendicular to the axis of the dipole and at a distance \(r\) from the center of the dipole, the electric field is:
\[ E_{\text{equatorial}} = \frac{1}{4 \pi \epsilon_0} \cdot \frac{p}{r^3} \]
### Key Relationships
1. **Distance Dependence**: In both cases, the electric field decreases with the cube of the distance from the dipole. This is a characteristic feature of the dipole field, as opposed to the \(1/r^2\) dependence seen in the field of a single point charge.
2. **Dipole Moment Influence**: The electric field also depends on the magnitude of the dipole moment \(p\). A larger dipole moment results in a stronger field at a given distance.
3. **Orientation Dependence**: The field strength also depends on the position relative to the dipole's orientation. The field is strongest along the axial line and weaker on the equatorial line.
### Summary
- **On the axial line**: The electric field \(E_{\text{axial}}\) is proportional to \( \frac{3p}{r^3} \).
- **On the equatorial line**: The electric field \(E_{\text{equatorial}}\) is proportional to \( \frac{p}{r^3} \).
In essence, the electric field of a short dipole decreases with the cube of the distance from the dipole, and its strength varies depending on the direction relative to the dipole's orientation.