When a capacitor is subjected to a D.C. source it takes a small time interval to get fully charged up. During this small-time interval, there is no passage of charge through the dielectric, yet we use a term - displacement current. This term is used because:
There is a continuous change of electric field between plates and hence electric flux
Understanding Displacement Current in Capacitor Charging
When a capacitor is connected to a DC source, it doesn't instantly reach its full charge. There is a time interval during which charge flows from the source to the capacitor plates, and the voltage across the capacitor builds up. A key concept during this process, especially in the region between the plates where the dielectric is, is displacement current.
Why Displacement Current is Used
Unlike the conduction current that flows through wires due to the movement of free charges, there is no flow of free charges through the ideal dielectric material between the capacitor plates. However, the question highlights the use of the term "displacement current". This term is crucial for the consistency of electromagnetic theory, particularly Maxwell's equations.
As the capacitor charges, charge accumulates on the plates. This accumulation of charge leads to the build-up of an electric field in the region between the plates. The electric field, \(\mathbf{E}\), is directly related to the charge density on the plates. As charge increases, the electric field strength between the plates also increases with time.
The electric flux, \(\Phi_E\), through any surface between the plates is given by the integral of the electric field over that surface. Since the electric field between the plates is changing with time, the electric flux through a surface in this region is also changing with time. Maxwell's correction to Ampère's law introduces the concept of displacement current, \(I_d\), which is proportional to the rate of change of electric flux:
\(I_d = \epsilon_0 \frac{d\Phi_E}{dt}\)
where \(\epsilon_0\) is the permittivity of free space (or \(\epsilon\) for a dielectric). This changing electric flux acts as a source of a magnetic field, just like a conduction current does.
Therefore, the term displacement current is used because the changing electric field (and consequently, the changing electric flux) between the plates during the charging process behaves electromagnetically like a current, even though no actual charge is moving through the dielectric.
Analyzing the Options
- Option 1: There is a slow motion of charge from one plate to another
This is incorrect. In a capacitor with a dielectric, ideally, no free charge moves through the dielectric from one plate to the other during normal operation (charging or discharging). Dielectrics are insulators. - Option 2: There is a continuous change of electric field between plates and hence electric flux
This statement accurately describes the situation during capacitor charging. As charge builds up on the plates, the electric field between the plates increases. A changing electric field results in a changing electric flux through any surface between the plates. This changing electric flux is the physical phenomenon that the displacement current term represents. - Option 3: The electric charge from the battery stops moving
This is incorrect. During the charging process, electric charge flows from the battery terminals, through the connecting wires (as conduction current), and accumulates on the capacitor plates. The flow stops only when the capacitor is fully charged to the battery voltage. - Option 4: There is a temporary breakdown of dielectric
This is incorrect. Dielectric breakdown is a failure mode where the electric field becomes strong enough to cause the dielectric to conduct, usually leading to damage. The concept of displacement current applies during the normal charging process, well before breakdown occurs.
Based on the analysis, the reason the term displacement current is used during capacitor charging is due to the continuous change in the electric field and electric flux between the plates.
| Concept | Description | Relevance to Capacitor Charging |
|---|---|---|
| Conduction Current | Flow of free electric charges (like electrons in wires). | Flows in the wires connecting the battery to the capacitor plates. |
| Displacement Current | Not a flow of charge, but a concept related to changing electric flux. | Exists in the region between the capacitor plates where the electric field and flux are changing. Necessary for the consistency of Maxwell's equations. |
| Dielectric | Insulating material between capacitor plates. | Prevents conduction current from flowing between plates, but permits the electric field to exist and change. |
| Electric Field (\(\mathbf{E}\)) | Force per unit charge. Exists between charged plates. | Increases as the capacitor charges, as more charge accumulates on the plates. |
| Electric Flux (\(\Phi_E\)) | Measure of the electric field passing through a given area. | Changes as the electric field between the plates changes. The rate of change of electric flux is proportional to the displacement current. |
Conclusion
The term displacement current is a theoretical construct introduced by James Clerk Maxwell. It arises from the time variation of the electric field. In the case of a charging capacitor, the electric field between the plates is constantly changing, leading to a changing electric flux. This changing flux is equivalent to a displacement current in terms of its ability to produce a magnetic field, thus ensuring that Ampère's law holds true even in regions where no conduction current exists, such as the space between the capacitor plates.
Revision Table: Capacitor Charging and Displacement Current
| Term | Definition | Why it matters here |
|---|---|---|
| Capacitor | Device storing electrical energy in an electric field. | Undergoes charging process, involving changing fields. |
| DC Source | Provides constant voltage/current over time. | Drives the charging of the capacitor. |
| Dielectric | Insulator between plates. | Blocks conduction current, allows electric field change. |
| Electric Field (E) | Exists between charged plates. | Changes as charge accumulates. |
| Electric Flux (\(\Phi_E\)) | Related to field lines passing through an area. | Changes as E changes, source of displacement current. |
| Displacement Current (\(I_d\)) | \(\epsilon_0 \frac{d\Phi_E}{dt}\) | Conceptually completes the circuit in Maxwell's theory; exists where flux changes. |
Additional Information: Maxwell's Equations and Currents
The concept of displacement current is a cornerstone of Maxwell's equations, which unify electricity and magnetism. Ampère's original law related the circulation of the magnetic field to the conduction current enclosed. Maxwell realized this law was incomplete when dealing with changing electric fields, such as in a charging capacitor.
Consider a charging capacitor. If you draw a loop enclosing one wire leading to the capacitor plate, Ampère's original law correctly predicts the magnetic field around the wire based on the conduction current. However, if you draw the same loop but place a surface through it that passes between the capacitor plates, there is no conduction current passing through this surface. Ampère's original law would predict no magnetic field, which contradicts experimental observation (there is a magnetic field between the plates!).
Maxwell resolved this by adding the displacement current term to Ampère's law. The corrected law, known as the Ampère-Maxwell law, is:
\(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 (I_c + I_d)\)
or in terms of electric flux:
\(\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_c + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}\)
Here, \(I_c\) is the conduction current and \(I_d = \epsilon_0 \frac{d\Phi_E}{dt}\) is the displacement current. This term ensures that the law is consistent for any surface bounded by the loop and correctly predicts the magnetic field between the capacitor plates, where \(I_c = 0\) but \(I_d \neq 0\) while charging. In the wire, \(I_c \neq 0\) but \(I_d = 0\) (assuming static fields outside the capacitor), and the total current (\(I_c + I_d\)) is continuous throughout the circuit during charging.
Important Questions from Electromagnetic Waves
When we draw the variation of the potential energy of a pair of nucleons with their separations, then:
A capacitor of 25μF is connected in series with a DC voltage of 5V. The value of current in the circuit will be:
Peak voltage of a modulating signal is 2 V. The carrier wave is represented by C(t) = 4sin(8πt)V. The modulation index of the modulated signal is:
A slab of material of dielectric constant k has the same area as the plates of a parallel plate capacitor, but has a thickness (3d/4), where d is the distance between plates of the capacitor. The ratio of the capacitance with the dielectric inside it to its capacitance without the dielectric is:
Arrange the following in increasing order of quantum number when coming from an excited energy state:
- A. Lyman Series
- B. Balmer Series
- C. Paschen Series
- D. Brackett Series
- E. Pfund Series
Choose the correct answer from the options given below:
