What is the relation between volume charge density and electric field intensity?
2 Answers
The relationship between volume charge density (\(\rho\)) and electric field intensity (\(\mathbf{E}\)) is fundamentally described by Gauss's Law, which is one of Maxwell's equations in electromagnetism. Here’s a detailed explanation of how they are related:
### 1. **Volume Charge Density (\(\rho\))**
Volume charge density is a measure of how much electric charge is distributed per unit volume at a point in space. It is defined mathematically as:
\[ \rho = \frac{dQ}{dV} \]
where \(dQ\) is the infinitesimal amount of charge in the infinitesimal volume \(dV\).
### 2. **Electric Field Intensity (\(\mathbf{E}\))**
Electric field intensity is a vector field that represents the force per unit charge exerted on a positive test charge placed in the field. It is defined as:
\[ \mathbf{E} = \frac{\mathbf{F}}{q} \]
where \(\mathbf{F}\) is the force experienced by the test charge \(q\).
### 3. **Gauss’s Law**
Gauss's Law relates the electric field to the volume charge density. It states that the electric flux through a closed surface is proportional to the enclosed charge. Mathematically, it’s expressed as:
\[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \]
where:
- \(\oint_S \mathbf{E} \cdot d\mathbf{A}\) is the electric flux through a closed surface \(S\),
- \(Q_{\text{enc}}\) is the total charge enclosed by the surface,
- \(\varepsilon_0\) is the permittivity of free space (a constant approximately equal to \(8.854 \times 10^{-12} \, \text{F/m}\)).
### 4. **Relating Charge Density to Electric Field**
To connect volume charge density to the electric field, consider a differential volume element in space. If we apply Gauss’s Law to a small volume element of the charge distribution, we get:
\[ \mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
where \(\mathbf{\nabla} \cdot \mathbf{E}\) is the divergence of the electric field, which measures how much the field spreads out from a point.
### 5. **Interpretation**
- **Divergence of Electric Field**: The divergence of the electric field vector \(\mathbf{E}\) at a point is directly proportional to the volume charge density at that point. If there is a positive charge density (\(\rho > 0\)), the field lines diverge from that point. If there is a negative charge density (\(\rho < 0\)), the field lines converge towards that point.
- **In Regions Without Charges**: In regions where there is no charge (\(\rho = 0\)), the divergence of the electric field is zero, which means the electric field lines neither converge nor diverge in that region.
### 6. **Example**
Consider a point charge \(Q\) located at the origin. For a spherically symmetric charge distribution, the electric field at a distance \(r\) from the origin is given by:
\[ \mathbf{E} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2} \hat{\mathbf{r}} \]
In this case, the volume charge density \(\rho\) is zero everywhere except at the origin, where it is mathematically infinite if considered as a point charge. The electric field follows from Gauss's Law, confirming that \(\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}\) holds true even in such idealized scenarios.
In summary, Gauss's Law provides a direct link between the volume charge density and the electric field intensity, showing that the divergence of the electric field is proportional to the volume charge density. This fundamental relationship helps in understanding how electric fields are generated by charge distributions and how they behave in different regions of space.
### 1. **Volume Charge Density (\(\rho\))**
Volume charge density is a measure of how much electric charge is distributed per unit volume at a point in space. It is defined mathematically as:
\[ \rho = \frac{dQ}{dV} \]
where \(dQ\) is the infinitesimal amount of charge in the infinitesimal volume \(dV\).
### 2. **Electric Field Intensity (\(\mathbf{E}\))**
Electric field intensity is a vector field that represents the force per unit charge exerted on a positive test charge placed in the field. It is defined as:
\[ \mathbf{E} = \frac{\mathbf{F}}{q} \]
where \(\mathbf{F}\) is the force experienced by the test charge \(q\).
### 3. **Gauss’s Law**
Gauss's Law relates the electric field to the volume charge density. It states that the electric flux through a closed surface is proportional to the enclosed charge. Mathematically, it’s expressed as:
\[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} \]
where:
- \(\oint_S \mathbf{E} \cdot d\mathbf{A}\) is the electric flux through a closed surface \(S\),
- \(Q_{\text{enc}}\) is the total charge enclosed by the surface,
- \(\varepsilon_0\) is the permittivity of free space (a constant approximately equal to \(8.854 \times 10^{-12} \, \text{F/m}\)).
### 4. **Relating Charge Density to Electric Field**
To connect volume charge density to the electric field, consider a differential volume element in space. If we apply Gauss’s Law to a small volume element of the charge distribution, we get:
\[ \mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
where \(\mathbf{\nabla} \cdot \mathbf{E}\) is the divergence of the electric field, which measures how much the field spreads out from a point.
### 5. **Interpretation**
- **Divergence of Electric Field**: The divergence of the electric field vector \(\mathbf{E}\) at a point is directly proportional to the volume charge density at that point. If there is a positive charge density (\(\rho > 0\)), the field lines diverge from that point. If there is a negative charge density (\(\rho < 0\)), the field lines converge towards that point.
- **In Regions Without Charges**: In regions where there is no charge (\(\rho = 0\)), the divergence of the electric field is zero, which means the electric field lines neither converge nor diverge in that region.
### 6. **Example**
Consider a point charge \(Q\) located at the origin. For a spherically symmetric charge distribution, the electric field at a distance \(r\) from the origin is given by:
\[ \mathbf{E} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2} \hat{\mathbf{r}} \]
In this case, the volume charge density \(\rho\) is zero everywhere except at the origin, where it is mathematically infinite if considered as a point charge. The electric field follows from Gauss's Law, confirming that \(\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}\) holds true even in such idealized scenarios.
In summary, Gauss's Law provides a direct link between the volume charge density and the electric field intensity, showing that the divergence of the electric field is proportional to the volume charge density. This fundamental relationship helps in understanding how electric fields are generated by charge distributions and how they behave in different regions of space.
The relationship between volume charge density (\(\rho\)) and electric field intensity (\(\mathbf{E}\)) is described by Gauss's Law, one of Maxwell's equations. Gauss's Law in its differential form relates the electric field to the charge distribution that produces it. Here’s how they are connected:
### Gauss's Law
In differential form, Gauss's Law states that:
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
where:
- \(\nabla \cdot \mathbf{E}\) is the divergence of the electric field \(\mathbf{E}\),
- \(\rho\) is the volume charge density,
- \(\varepsilon_0\) is the permittivity of free space.
### Explanation
1. **Volume Charge Density (\(\rho\))**: This is a measure of how much electric charge is distributed within a given volume. It is usually expressed in units of coulombs per cubic meter (\(\text{C/m}^3\)).
2. **Electric Field Intensity (\(\mathbf{E}\))**: This is a vector field that represents the force per unit charge experienced by a positive test charge placed at a point in space.
3. **Divergence of the Electric Field**: The divergence of the electric field (\(\nabla \cdot \mathbf{E}\)) at a point in space measures how much the electric field is "spreading out" from or converging into that point. It essentially tells us about the presence of a source or sink of electric field lines.
### Interpretation
- **Positive Volume Charge Density**: If \(\rho > 0\), there is a net positive charge density, and the divergence of the electric field \(\nabla \cdot \mathbf{E}\) will be positive. This indicates that the electric field lines are diverging from the positive charge.
- **Negative Volume Charge Density**: If \(\rho < 0\), there is a net negative charge density, and the divergence of the electric field \(\nabla \cdot \mathbf{E}\) will be negative. This indicates that the electric field lines are converging towards the negative charge.
- **Zero Volume Charge Density**: If \(\rho = 0\), the divergence of the electric field is also zero, which implies that there are no sources or sinks of the electric field in that region.
In summary, Gauss's Law connects the electric field intensity and volume charge density by stating that the divergence of the electric field at a point is proportional to the local volume charge density at that point. This provides a direct link between the distribution of electric charge and the resulting electric field in space.
### Gauss's Law
In differential form, Gauss's Law states that:
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
where:
- \(\nabla \cdot \mathbf{E}\) is the divergence of the electric field \(\mathbf{E}\),
- \(\rho\) is the volume charge density,
- \(\varepsilon_0\) is the permittivity of free space.
### Explanation
1. **Volume Charge Density (\(\rho\))**: This is a measure of how much electric charge is distributed within a given volume. It is usually expressed in units of coulombs per cubic meter (\(\text{C/m}^3\)).
2. **Electric Field Intensity (\(\mathbf{E}\))**: This is a vector field that represents the force per unit charge experienced by a positive test charge placed at a point in space.
3. **Divergence of the Electric Field**: The divergence of the electric field (\(\nabla \cdot \mathbf{E}\)) at a point in space measures how much the electric field is "spreading out" from or converging into that point. It essentially tells us about the presence of a source or sink of electric field lines.
### Interpretation
- **Positive Volume Charge Density**: If \(\rho > 0\), there is a net positive charge density, and the divergence of the electric field \(\nabla \cdot \mathbf{E}\) will be positive. This indicates that the electric field lines are diverging from the positive charge.
- **Negative Volume Charge Density**: If \(\rho < 0\), there is a net negative charge density, and the divergence of the electric field \(\nabla \cdot \mathbf{E}\) will be negative. This indicates that the electric field lines are converging towards the negative charge.
- **Zero Volume Charge Density**: If \(\rho = 0\), the divergence of the electric field is also zero, which implies that there are no sources or sinks of the electric field in that region.
In summary, Gauss's Law connects the electric field intensity and volume charge density by stating that the divergence of the electric field at a point is proportional to the local volume charge density at that point. This provides a direct link between the distribution of electric charge and the resulting electric field in space.