We first show why the electric charge in an LC circuit oscillates. We then use the laws governing electric and magnetic fields to determine how electromagnetic waves propagate through a vacuum and how fast they travel.
Adjust conditions:
The sliders adjust the values
電荷が初めの半分になった瞬間、電流の大きさは最大値の何倍か？
C=1.0 µFで抵抗63.2 Ωを加えると、電荷の符号は反転するか？
The equations explain the phenomenon
1The electric potential differences across the coil and capacitor give the circuit equation.L dIdt=−QC……
2Substituting I = dQ/dt allows comparison with the spring equation.d²Qdt²=−QL C……
3The initial conditions determine the expressions for electric charge and electric current.Q=Q₀ cos ωt，I=−Q₀ω sin ωt……
4Multiplying the circuit equation by I gives conservation of energy.½ Q²C+½ L I²=½ Q₀²C……
5The electric current can be calculated when the electric charge is halved.|I|Imax=√32……
6Add resistance and check whether the electric charge changes sign.L d²Qdt²+R dQdt+QC=0……
7Values from a radio circuit give its oscillation frequency.f=12π √(L C)……
8An additional law connects changing electric and magnetic fields.∮ B · dl=μ₀ I+μ₀ ε₀ dΦEdt……
9Changes in electric and magnetic fields propagate as waves.E ⊥ B，E ⊥ direction of propagation，B ⊥ direction of propagation……
10The electromagnetic wave speed is calculated from ε₀ and μ₀.c=1√(ε₀ μ₀)……
Further exploration: wavelengths and familiar electromagnetic waves
11Electromagnetic waves are arranged by wavelength.λ=cf……
What this chapter has shown
When losses due to electrical resistance and radiation are neglected, the electric charge in an LC circuit undergoes simple harmonic motion governed by the same equation as a spring, with energy alternating between the electric and magnetic fields. In a vacuum, Maxwell's equations relate the changes in the electric and magnetic fields over time and space. These equations have solutions that propagate as waves at a speed of 1/√(ε₀μ₀), which equals the speed of light.
0.866 timesCurrent ratio when the electric charge falls to half its initial value
If the initial electric current is 0 and the electrical resistance is at least 2√(L/C), the electric charge approaches 0 without changing sign.
In everyday lifeA radio adjusts C to match the circuit's frequency to that of a station's radio waves, making the circuit respond strongly to those waves.
Detailed summary
A circuit containing a capacitor and an inductor oscillates according to the same equation as a spring system, transferring energy between the electric and magnetic fields. The electric charge and electric current do not reach their maximum values at the same time. When the charge is half its maximum value, the current is √3/2 times its maximum value. Combining the 2 laws describing how a changing magnetic field produces an electric field and how a changing electric field produces a magnetic field shows that the electric and magnetic fields satisfy a wave equation and propagate through a vacuum as an electromagnetic wave. Its speed is determined solely by the electric and magnetic constants and was found to match the speed of light.