**Energy Stored in a Magnetic Field**

When we pull two particles with opposite signs of charge away from each other, we say that the resulting electric potential energy is stored in the electric field of the particles. We get it back from the field by letting the particles move closer together again. In the same way we can consider energy to be stored in a magnetic field. To derive a quantitative expression for that stored energy, consider again, which shows a source of emf ‘(connect to a resistor R and an inductor L. Equation 31-41. restated here for convenience is the differential equation that describes the growth of current in this circuit. Recall that this equation follows immediately from the loop rule and that the loop rule in turn is an expression of the principle of conservation of energy for single-loop circuits. If we multiply each side of by I,

1. If a charge dq passes through the battery of emf ‘(gin Fig. 31-18 in time dt, the battery does work on it in the amount ‘(gdq. The rate at which the battery does work is (‘(g dq)/tit, or ‘(gj. Thus, the left side of Eq. 31-49 represents the rate at which the emf device delivers energy to the rest of the circuit.

2. he rightmost term in Eq. 31-49 represents the rate at which energy appears as thermal energy in the resistor.

3. Energy that is delivered to the circuit but does not appear as thermal energy by the conservation-of-energy hypothesis. be stored in the magnetic field of the inductor. Since Eq. 31-49 represents the principle of conservation of energy for RL circuits, the middle term must represent the rate dUB/dt at which energy isstored in the magnetic field.

This equation gives the density of stored energy at any point where the magnetic field is B. Even though we derived it by considering the special case of a solenoid, Eq. 3 I -56 holds for all magnetic fields, no matter how they are generated. The equation is comparable to Eq. 26-23; namely,

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