Two parallel metal plates are charged, and a dielectric is slowly inserted between them. The cases with the battery disconnected and with the battery connected are compared.
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電池を外して板間全体を比誘電率4の誘電体で満たすと、エネルギーは何倍か？
電池につないだまま同じ誘電体を入れると、蓄えられるエネルギーは何倍か？
The equations explain the phenomenon
1An electric charge moves in an electric field.W=q E d……
2Electric potential is defined as potential energy per 1 C of electric charge.U=q φ……
3Electric potential difference is related to the electric field.V=E d……
4The electric potential around a point charge is calculated.φ=k Qr……
5Points with the same electric potential are connected.E⊥Equipotential lines……
6A voltage is applied across 2 plates.Q=C V，C=ε₀ Sd……
7A dielectric is inserted between the plates, and the capacitance is calculated.C'=εr ε₀ Sd……
From this point on, U represents the total energy stored in the capacitor. In step 2, U = qφ represents the potential energy of a test charge. The electric charge during charging is denoted by q, and the final electric charge by Q.
8The stored energy is calculated from the work required to charge the capacitor.U=½ Q V=½ C V²……
9The dielectric is inserted after the battery is disconnected.U'=Q²2 εr C=Uεr……
10The dielectric is inserted with the battery connected.U'=½ εr C V²=εr U……
Further study: equivalent capacitance and electric field energy
11Capacitors are connected together.In parallel: C₁+C₂，In series: 11/C₁ + 1/C₂……
12Energy is expressed in terms of the electric field.u=½ ε₀ E²……
What this chapter has shown
Inserting a dielectric increases the capacitance C. With the battery disconnected, Q is constant, so V=Q/C decreases and the stored energy U=½CV² decreases. With the battery connected, V is constant, so Q=CV increases and the stored energy also increases. In both cases, the dielectric raises the weight as it is pulled inward.
0.25 times the initial valueU' after insertion/U before insertion
With the battery connected, inserting the same dielectric instead increases the energy by a factor of 4.
In everyday lifeA camera flash produces light by rapidly discharging the electric charge stored in a capacitor.
Detailed summary
The work done by the electrostatic force is independent of the path, so potential energy can be defined for it, just as for gravity. Electric potential φ is the potential energy per 1 C of charge, and the electric field describes how steeply the electric potential changes with position. A capacitor stores electric charge in proportion to the voltage across it and holds energy ½CV² in the electric field between its plates. Whether inserting a dielectric increases or decreases the energy depends on whether the charge or the voltage is held constant.