Discharging a Capacitor: Magnetic Fields ⚡
What Happens When a Capacitor Discharges?
Think of a capacitor as a water tank filled with electric charge. When you open the valve (connect a resistor), the water (charge) starts flowing out. The flow of charge is called current and it creates a magnetic field, just like a running river creates ripples.
Key Equations for an RC Discharge
- Voltage across the capacitor: \$V(t) = V_0 e^{-t/RC}\$
- Current in the resistor: \$I(t) = \dfrac{V_0}{R} e^{-t/RC}\$
- Magnetic field around a long straight wire (Biot‑Savart): \$B(r,t) = \dfrac{\mu_0 I(t)}{2\pi r}\$
Analogy: The River of Current
Imagine the discharge current as a river flowing through a straight pipe. The magnetic field lines are like concentric circles around the pipe, similar to how ripples spread out from a stone dropped in water. The faster the current, the stronger the ripples (magnetic field).
Exam Tip: Remember the Relationship
Key point: The magnetic field is directly proportional to the instantaneous current. As the capacitor discharges, the current decreases exponentially, so the magnetic field also decays exponentially.
When answering exam questions, always:
- Identify the circuit configuration (series RC).
- Write down the expression for \$I(t)\$.
- Use \$B(r,t) = \dfrac{\mu_0 I(t)}{2\pi r}\$ for a long straight wire.
- Explain the direction of \$B\$ using the right‑hand rule.
Example Problem
Given a 10 µF capacitor charged to 5 V and a 1 kΩ resistor, calculate the magnetic field at a distance of 2 cm from the wire after 5 ms.
- Compute the time constant: \$\tau = RC = (10\times10^{-6}\,\text{F})(1000\,\Omega) = 0.01\,\text{s}\$.
- Find \$I(5\,\text{ms}) = \dfrac{5\,\text{V}}{1000\,\Omega} e^{-0.005/0.01} \approx 5\,\text{mA}\times e^{-0.5} \approx 3.05\,\text{mA}\$.
- Calculate \$B(2\,\text{cm}) = \dfrac{(4\pi\times10^{-7}\,\text{H/m})(3.05\times10^{-3}\,\text{A})}{2\pi(0.02\,\text{m})} \approx 9.7\times10^{-6}\,\text{T}\$.
| Time (ms) | Current I(t) (mA) | Magnetic Field B(2 cm) (µT) |
|---|
| 0 | 5.00 | 15.9 |
| 5 | 3.05 | 9.7 |
| 10 | 1.85 | 5.9 |
Quick Review for Exams
- Discharge current decays exponentially: \$I(t) = I_0 e^{-t/RC}\$.
- Magnetic field around a straight wire: \$B = \dfrac{\mu_0 I}{2\pi r}\$.
- Direction of \$B\$ given by the right‑hand rule (thumb along current, fingers curl).
- Remember that \$B\$ also decays exponentially with time as \$I(t)\$ does.