Energy density in physics is calculated as the amount of energy stored in a system or region of space per unit volume. It represents how concentrated the energy is.
Here's how you can calculate it, generally and in specific contexts:
General Formula
The general formula for energy density (usually denoted by the symbol u) is:
u = U / V
where:
- u is the energy density
- U is the total energy stored in the system
- V is the volume over which the energy is distributed
Think of it as energy divided by volume, much like mass density is mass divided by volume.
Energy Density in a Capacitor
As illustrated in the reference video, to calculate the energy density within a capacitor:
- Determine the total energy stored in the capacitor (U). This can be calculated using the formula:
U = (1/2) * C * V^2, where C is the capacitance and V is the voltage. - Determine the volume (V) of the space where the energy is stored. For a parallel-plate capacitor, this is typically the area of the plates multiplied by the distance between them (i.e.,
V = A * d). - Divide the total energy (U) by the volume (V). This gives you the energy density (u):
u = U / V.
Therefore, u = (1/2) * C * V^2 / (A * d)
It can be shown that this simplifies to u = (1/2) * ε₀ * E^2, where ε₀ is the permittivity of free space and E is the electric field. This form is especially useful because it expresses energy density in terms of a local property (the electric field) instead of macroscopic properties like capacitance and volume.
Energy Density of an Electric Field
The energy density associated with an electric field is given by:
u = (1/2)ε₀E²
where:
- u is the energy density
- ε₀ is the permittivity of free space (approximately 8.854 × 10⁻¹² F/m)
- E is the magnitude of the electric field
Energy Density of a Magnetic Field
Similarly, the energy density associated with a magnetic field is given by:
u = (1/2)(B²/μ₀)
where:
- u is the energy density
- B is the magnitude of the magnetic field
- μ₀ is the permeability of free space (approximately 4π × 10⁻⁷ H/m)
Example
Let's say you have a region with an electric field strength of 1000 V/m. The energy density in that region would be:
u = (1/2) (8.854 × 10⁻¹² F/m) (1000 V/m)²
u = 4.427 × 10⁻⁶ J/m³
This means that every cubic meter in that region stores 4.427 × 10⁻⁶ joules of energy in the electric field.
Summary
Calculating energy density involves dividing the total energy stored in a system by the volume it occupies. Depending on the situation (capacitor, electric field, magnetic field), specific formulas are used, but the fundamental principle remains the same: determining the concentration of energy in a given space.
