Energy Density and Electric Energy Density Formulas with Examples
What is Energy Density?
Energy Density refers to the total amount of energy in a system per unit volume. (Even though generally energy per unit mass is also mentioned as energy density, the proper term for the same is specific energy. The term density usually measures the amount per unit spatial extension). Hence energy density is stored energy in a given system or region of space per unit volume. In this article, we will learn about the formula of calculation of the energy density of any system.
Energy Density Formula
Energy density is denoted by the letter U.
Magnetic and electric fields can also store energy.
In the case of an electric field or capacitor, the energy density is given by UE = \[\frac{1}{2}\]εoE2
The energy density in the case of magnetic field or inductor is given by, UB =\[\frac{1}{2\mu _0}\] B2
Where UE = Electrical energy density
UB= Magnetic energy density
εo=Permittivity
E= Electric field
B=Magnetic field
μ= magnetic permeability
For electromagnetic waves, both magnetic and electric fields are equally involved in contributing to energy density. Therefore, the energy density is the sum of the energy density of electric and magnetic fields.
i.e., U = \[\frac{1}{2}\]εoE2 + \[\frac{1}{2\mu _0}\]B2
Solved Examples
Example 1: In a certain region of space, the magnetic field has a value of 1.0 × 10–2 T, and the electric field has a value of 2.0 ×106 Vm-1. Find the combined energy density of the electric and magnetic fields.
Solution: E = 2.0 × 106 Vm-1; B = 1.0 × 10-2 T
For the electric field, the energy density is UE = \[\frac{1}{2}\] εoE2
=12 × 8.85 × 10−12(2.0 × 106)2 = 18Jm−3
For the magnetic field, the energy density is UB =\[\frac{1}{2\mu _0}\]B2
=\[\frac{1}{2} \times \frac{(1.0 \times 10^{-2})^{2}}{4\pi \times 10^{-7}} = 40jm^{-3}\]
The net energy density is the sum of the energy density due to the electric field and the energy density due to the magnetic field:
U= UE + UB = 18 + 40 = 58 Jm-3
Example 2: In a certain region of space, the magnetic field has a value of 3X10-2T, and the electric field has a value of 9X107 V/m. Calculate the energy density of the electric and magnetic fields?
Solution: B = 3 X 10-2T, E = 9 X 107 V/m, ε = 8.85 X 10-12 C2 / Nm2 and μ = 4π X 10-7 N/A2.
UE = 12εoE2
UE = 8.85 X 10-12 C2 / Nm2 X (9X107V/m)2/2
UE = 35842.5 J/m3
UB =12μ0B2
UB = (3 X 10-2T)2 / 2 X 4π X 10-7 N/A2
UB = 358.1J/m3
U = UE + UB
U = 35,842.5 J/m3 + 358.1 J/m3
U = 36200.6 J/m3
Conclusion
The formula of the energy density of a system is the sum of the energy density of electric fields and magnetic fields. The above article is useful for the students to revise the formula of energy density and the solved example will help them to understand how to apply the formula in numerical.
FAQs on Energy Density Formula
1. Why does the energy density formula the sum of the energy density of electric and magnetic fields?
Energy density is defined as the total amount of energy stored in a system per unit volume. The main sources of energy or energy density are magnetic fields and electric fields. We know that both magnetic and electric fields are involved in contributing to energy density equally regarding the electromagnetic waves. Therefore the formula of energy density is the sum of the energy density of electric and magnetic fields both together.
2. How to calculate Energy Density?
Energy is brought by electromagnetic waves into any system by the electric field and magnetic field. The charges can do movement because these fields exert forces in the system and thus work is done on them. The energy density of an electromagnetic wave can be calculated with help of the formula of energy density which is U = \[\frac{1}{2} \epsilon _oE^2 + \frac{1}{2\mu _0} B^2\]. The formula is the sum of the energy density of electric and magnetic fields.