Solved Examples of Magnetic Energy Density Calculations

📚 Quick Study Guide

• 🧲 Magnetic energy density ($u_B$) is the energy stored per unit volume in a magnetic field.
• 📐 The formula for magnetic energy density is: $u_B = \frac{B^2}{2\mu_0}$, where $B$ is the magnetic field strength and $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7} \text{ T m/A}$).
• ⚡ The total magnetic energy ($U_B$) stored in a volume is found by integrating the magnetic energy density over the volume: $U_B = \int u_B dV$.
• 💡 For a solenoid, the magnetic field inside is approximately uniform, allowing for simpler calculations of total energy.
• 🔢 Remember to use consistent units (Tesla for magnetic field, meters for length, and Amperes for current).

Practice Quiz

1. A region has a uniform magnetic field of 0.5 T. What is the magnetic energy density in this region?
   1. 0.099 J/m³
   2. 99.5 J/m³
   3. 0.99 J/m³
   4. 994.7 J/m³
2. A solenoid of length 0.2 m has a magnetic field of 2 T inside. What is the magnetic energy density inside the solenoid?
   1. 1.59 x 10⁶ J/m³
   2. 1.59 x 10³ J/m³
   3. 1.59 x 10⁴ J/m³
   4. 1.59 x 10⁵ J/m³
3. If the magnetic energy density in a certain region is 1000 J/m³, what is the magnetic field strength in that region?
   1. 1.58 T
   2. 0.00158 T
   3. 0.079 T
   4. 0.000079 T
4. A volume of 0.1 m³ has a uniform magnetic energy density of 500 J/m³. What is the total magnetic energy stored in this volume?
   1. 50 J
   2. 5 J
   3. 500 J
   4. 0.05 J
5. The magnetic field in a region is given by $B = 0.1x$ T, where $x$ is in meters. What is the magnetic energy density at $x = 2$ m?
   1. 1.59 J/m³
   2. 0.159 J/m³
   3. 0.0159 J/m³
   4. 15.9 J/m³
6. A toroid has a magnetic field of 0.8 T inside. Determine the magnetic energy density within the toroid.
   1. 2.55 x 10⁵ J/m³
   2. 2.55 x 10² J/m³
   3. 2.55 x 10³ J/m³
   4. 2.55 x 10⁴ J/m³
7. If the magnetic energy stored in a 0.02 m³ volume is 100 J, what is the average magnetic energy density in that volume?
   1. 500 J/m³
   2. 5000 J/m³
   3. 200 J/m³
   4. 2000 J/m³