What is the relation between current density and electric field in vector form?
2 Answers
The relation between current density \(\mathbf{J}\) and the electric field \(\mathbf{E}\) in vector form is given by Ohm's law for a conductive material. In vector form, this relationship can be expressed as:
\[ \mathbf{J} = \sigma \mathbf{E} \]
where:
- \(\mathbf{J}\) is the current density vector.
- \(\sigma\) is the electrical conductivity of the material.
- \(\mathbf{E}\) is the electric field vector.
### Explanation:
1. **Current Density (\(\mathbf{J}\))**: This vector quantity represents the amount of electric current flowing per unit area of a cross-section through a material. It has units of amperes per square meter (A/m²).
2. **Electric Field (\(\mathbf{E}\))**: This vector quantity represents the force per unit charge exerted on a charged particle in the material. It has units of volts per meter (V/m).
3. **Electrical Conductivity (\(\sigma\))**: This scalar quantity is a measure of a material's ability to conduct an electric current. It is the reciprocal of resistivity and has units of siemens per meter (S/m).
### In Summary:
The relationship \(\mathbf{J} = \sigma \mathbf{E}\) states that the current density \(\mathbf{J}\) is directly proportional to the electric field \(\mathbf{E}\), with the proportionality constant being the electrical conductivity \(\sigma\). This equation is fundamental in describing how electric currents flow through materials and is a key principle in electrical engineering and physics.
\[ \mathbf{J} = \sigma \mathbf{E} \]
where:
- \(\mathbf{J}\) is the current density vector.
- \(\sigma\) is the electrical conductivity of the material.
- \(\mathbf{E}\) is the electric field vector.
### Explanation:
1. **Current Density (\(\mathbf{J}\))**: This vector quantity represents the amount of electric current flowing per unit area of a cross-section through a material. It has units of amperes per square meter (A/m²).
2. **Electric Field (\(\mathbf{E}\))**: This vector quantity represents the force per unit charge exerted on a charged particle in the material. It has units of volts per meter (V/m).
3. **Electrical Conductivity (\(\sigma\))**: This scalar quantity is a measure of a material's ability to conduct an electric current. It is the reciprocal of resistivity and has units of siemens per meter (S/m).
### In Summary:
The relationship \(\mathbf{J} = \sigma \mathbf{E}\) states that the current density \(\mathbf{J}\) is directly proportional to the electric field \(\mathbf{E}\), with the proportionality constant being the electrical conductivity \(\sigma\). This equation is fundamental in describing how electric currents flow through materials and is a key principle in electrical engineering and physics.
The relationship between current density \(\mathbf{J}\) and electric field \(\mathbf{E}\) in vector form is given by Ohm's Law for a conductor. In its vectorial form, Ohm's Law is expressed as:
\[ \mathbf{J} = \sigma \mathbf{E} \]
Here's a detailed breakdown of the terms and concepts involved:
1. **Current Density (\(\mathbf{J}\))**: This is a vector quantity that represents the amount of electric current flowing per unit area through a given surface. It points in the direction of the current flow and has units of amperes per square meter (A/m²).
2. **Electric Field (\(\mathbf{E}\))**: This is also a vector quantity that represents the force per unit charge experienced by a charged particle in an electric field. It points in the direction that a positive test charge would be pushed and has units of volts per meter (V/m).
3. **Electrical Conductivity (\(\sigma\))**: This is a scalar quantity that measures how easily a material allows electric current to flow through it. It is the reciprocal of resistivity (\(\rho\)), where \(\sigma = 1/\rho\). Conductivity has units of siemens per meter (S/m).
### Interpretation
In the relation \(\mathbf{J} = \sigma \mathbf{E}\):
- **Directionality**: The direction of the current density vector \(\mathbf{J}\) is the same as the direction of the electric field vector \(\mathbf{E}\). This implies that in a conductive material, the current flows in the direction of the electric field.
- **Magnitude**: The magnitude of the current density is directly proportional to the magnitude of the electric field. The proportionality constant is the electrical conductivity \(\sigma\). A higher conductivity means that a given electric field will result in a larger current density.
### Physical Meaning
This relationship reflects how a material responds to an electric field. In a good conductor (like metals), \(\sigma\) is large, so even a small electric field will induce a significant current. In insulators, \(\sigma\) is very small, so even a strong electric field results in a very small current.
### In Different Contexts
- **Ohmic Materials**: For materials where \(\sigma\) is constant and the relationship holds true under varying electric fields, the material is said to obey Ohm's Law. These are called ohmic materials.
- **Non-Ohmic Materials**: In some materials, \(\sigma\) may depend on the electric field or current density, leading to a nonlinear relationship. These materials do not follow Ohm’s Law strictly.
To summarize, the vector relationship \(\mathbf{J} = \sigma \mathbf{E}\) encapsulates how electric currents are generated by electric fields in conductive materials, with the current density being proportional to the electric field strength and the proportionality constant being the material's conductivity.
\[ \mathbf{J} = \sigma \mathbf{E} \]
Here's a detailed breakdown of the terms and concepts involved:
1. **Current Density (\(\mathbf{J}\))**: This is a vector quantity that represents the amount of electric current flowing per unit area through a given surface. It points in the direction of the current flow and has units of amperes per square meter (A/m²).
2. **Electric Field (\(\mathbf{E}\))**: This is also a vector quantity that represents the force per unit charge experienced by a charged particle in an electric field. It points in the direction that a positive test charge would be pushed and has units of volts per meter (V/m).
3. **Electrical Conductivity (\(\sigma\))**: This is a scalar quantity that measures how easily a material allows electric current to flow through it. It is the reciprocal of resistivity (\(\rho\)), where \(\sigma = 1/\rho\). Conductivity has units of siemens per meter (S/m).
### Interpretation
In the relation \(\mathbf{J} = \sigma \mathbf{E}\):
- **Directionality**: The direction of the current density vector \(\mathbf{J}\) is the same as the direction of the electric field vector \(\mathbf{E}\). This implies that in a conductive material, the current flows in the direction of the electric field.
- **Magnitude**: The magnitude of the current density is directly proportional to the magnitude of the electric field. The proportionality constant is the electrical conductivity \(\sigma\). A higher conductivity means that a given electric field will result in a larger current density.
### Physical Meaning
This relationship reflects how a material responds to an electric field. In a good conductor (like metals), \(\sigma\) is large, so even a small electric field will induce a significant current. In insulators, \(\sigma\) is very small, so even a strong electric field results in a very small current.
### In Different Contexts
- **Ohmic Materials**: For materials where \(\sigma\) is constant and the relationship holds true under varying electric fields, the material is said to obey Ohm's Law. These are called ohmic materials.
- **Non-Ohmic Materials**: In some materials, \(\sigma\) may depend on the electric field or current density, leading to a nonlinear relationship. These materials do not follow Ohm’s Law strictly.
To summarize, the vector relationship \(\mathbf{J} = \sigma \mathbf{E}\) encapsulates how electric currents are generated by electric fields in conductive materials, with the current density being proportional to the electric field strength and the proportionality constant being the material's conductivity.