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Understanding Capacitance
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Let's begin with our first exercise about calculating capacitance. How do we determine the capacitance of a capacitor?
Isn't it related to the size of the plates and the distance between them?
Exactly! The capacitance C is given by the formula C = Ξ΅0 * (A/d), where A is the area of the plates and d is the separation.
What if the plates are circular like in our exercise?
Great question! For circular plates, the area A would be ΟrΒ². So, you would plug that into the formula. Can anyone tell me what the capacitance would be if the radius is 12 cm and a distance of 5 cm?
I think weβd first find the area using A = Ο * (0.12 m)Β² and then calculate it!
Correct! Remember to convert centimeters to meters! Let's summarize: Capacitance depends on plate area and separation.
Exploring Displacement Current
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Moving to the next exercise, who can explain what displacement current is?
Itβs related to changing electric fields, right? Like in a charging capacitor?
Exactly! Displacement current is given by i_d = Ρ0 * (dΦ_E/dt). In our exercises, you will calculate it based on the charging current. Can you point out the steps?
First, we need the rate of change of electric flux, which comes from knowing the capacitance and voltage.
Spot on! This understanding will help you handle questions involving changes over time effectively.
Analyzing Kirchoff's Rules
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Letβs analyze Kirchhoff's Junction Rule for the capacitor examples. How does it apply here?
I think it states that total current entering a junction is equal to total current leaving?
Correct! Since we have both conduction and displacement currents, we need to consider both when analyzing the plates of the capacitor. What happens if we use different surfaces for calculation?
We might get different results unless we account for all currents!
Exactly! Great teamwork! Let's summarize: Kirchhoff's laws help maintain consistency in current calculations across different capacitor scenarios.
Relating Electric and Magnetic Fields
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Now, letβs dive into electromagnetic waves. How are the electric and magnetic fields related in these waves?
They are perpendicular to each other and to the direction of propagation!
Precisely! This relationship is crucial for the exercises. If I say the electric field amplitude is E, what's the amplitude of the magnetic field?
Is it B = E/c?
Yes! Where c is the speed of light. Letβs summarize this key takeaway: Electric and magnetic fields oscillate perpendicular to the direction of wave propagation.
Introduction & Overview
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Quick Overview
Standard
In this section, students are provided with exercises that challenge their understanding of key concepts related to electromagnetic waves and displacement current in circuits. The exercises encourage critical thinking and problem-solving regarding capacitor behavior, current types, wave properties, and electromagnetic spectrum characteristics.
Detailed
Exercises in Electromagnetism
This section includes various exercises designed to reinforce understanding of electromagnetic principles, particularly focusing on displacement current, capacitors, and electromagnetic waves. Each exercise encourages practical application of theoretical knowledge. The problems cover different levels of difficulty, allowing students to engage with the material at their own pace.
Key Concepts Covered
- Capacitance and Voltage: Understanding how to calculate capacitance and the relationship between voltage and charge in a capacitor.
- Displacement Current: Application of formulas to find the displacement current in circuits with alternating currents.
- Kirchhoff's Laws: Understanding the application of Kirchhoffβs rules to circuits with capacitors.
- Electromagnetic Wave Properties: Calculating magnetic fields generated by electromagnetic waves and understanding the relationships among electric and magnetic fields.
The exercises serve not only as a review but also challenge the students to apply learned concepts creatively and analytically.
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Audio Book
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Exercise 8.1: Capacitor Calculation
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Figure 8.5 shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15A.
(a) Calculate the capacitance and the rate of change of potential difference between the plates.
(b) Obtain the displacement current across the plates.
(c) Is Kirchhoffβs first rule (junction rule) valid at each plate of the capacitor? Explain.
Detailed Explanation
This exercise involves calculating the capacitance of a capacitor and understanding the displacement current. A capacitor stores electrical energy in the electric field between its plates. The capacitance (C) can be determined using the formula C = Ξ΅β(A/d), where A is the area of the plates and d is the distance between them. The displacement current can be computed using the changing electric field, which relates to the same current flowing through the capacitor. Kirchhoffβs junction rule states that the total current entering a junction must equal the total current leaving, which must be analyzed at the capacitor plates considering both conduction and displacement currents.
Examples & Analogies
Imagine a water tank where the incoming water flow represents the current. The capacity of the tank reflects the capacitance while the flow rate at any point in time corresponds to the changing potential difference. Just like how you can have water flowing in while it is being stored, charge accumulates in a capacitor while the current continues to flow.
Exercise 8.2: Parallel Plate Capacitor
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A parallel plate capacitor (Fig. 8.6) made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad sβ1.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of B at a point 3.0 cm from the axis between the plates.
Detailed Explanation
In this exercise, we analyze a parallel plate capacitor connected to an alternating current (AC) supply. The root mean square (RMS) value of the conduction current can be calculated using the formula I = C * V * Ο, where Ο is the angular frequency. We need to check if the conduction current equals the displacement current by understanding the relationship established by Maxwell. Finally, knowing the electric field (E) can help determine the magnetic field amplitude (B) at a given distance using the relationship B = E/c.
Examples & Analogies
Consider a hose filling a balloon with water where the hose's diameter (like capacitance) determines how quickly water can fill the balloon (representing the current). Just as the water pressure (like voltage) varies in an AC system, it affects how rapidly the balloon fills up. Meanwhile, if the balloon's surface expands and contracts with the water flow, this can model how displacement currents occur in conjunction with conduction currents.
Exercise 8.3: X-rays and Other Waves
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What physical quantity is the same for X-rays of wavelength 10β10 m, red light of wavelength 6800 Γ and radiowaves of wavelength 500m?
Detailed Explanation
This exercise prompts students to think about a fundamental property of electromagnetic waves: all waves, regardless of their type or wavelength, travel at the speed of light in vacuum. Thus, no matter if they are X-rays, visible light, or radio waves, their behavior under vacuum circumstances will be consistent, with each type having its own unique applications and energy characteristics.
Examples & Analogies
Think of how different sizes of cars (X-ray, visible light, radio waves) travel on the same road (vacuum), and while their sizes and models differβjust as wavelengths and frequencies differβthe speed limit (speed of light) applies to all equally.
Exercise 8.4: Plane Electromagnetic Wave
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A plane electromagnetic wave travels in vacuum along z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is 30 MHz, what is its wavelength?
Detailed Explanation
In a plane electromagnetic wave, the electric field (E) and the magnetic field (B) are always perpendicular to each other and to the direction of propagation of the wave. The wavelength (Ξ») can be computed using the relationship c = fΞ», where c is the speed of light and f is the frequency. For a frequency of 30 MHz, the corresponding wavelength can be calculated as Ξ» = c/f.
Examples & Analogies
Imagine standing on a beach where the waves approach the shore (propagation direction). The splashes you see in front (electric field) are in a different plane than the waves rolling toward you (magnetic field), much like how waves interact in higher-dimensional space.
Exercise 8.5: Radio Station Wavelength
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A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band?
Detailed Explanation
To find the wavelength band for radio frequencies, we can convert frequencies to wavelengths using the formula Ξ» = c/f. By inputting the frequencies for the range provided, we can calculate the minimum and maximum wavelengths for this band of radio frequencies.
Examples & Analogies
Think of tuning into a radio like looking for different colors in a rainbow (the wavelengths). Each frequency corresponds to a shade, and as you adjust the radio (like moving your vision), you shift through the spectrum, finding various shades which are the 'reports' from radio stations.
Exercise 8.6: Frequency of Oscillation
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A charged particle oscillates about its mean equilibrium position with a frequency of 10^9 Hz. What is the frequency of the electromagnetic waves produced by the oscillator?
Detailed Explanation
In this case, the frequency of the oscillating charge directly relates to the frequency of the electromagnetic waves it produces. Therefore, if the charged particle oscillates at 10^9 Hz, it emits electromagnetic waves of the same frequency.
Examples & Analogies
Similar to a child on a swing. If they oscillate back and forth at a certain rhythm (frequency), the waves produced by the movement (the air surrounding) can help visualize this processβjust like the electromagnetic radiation surrounding the charge.
Exercise 8.7: Magnetic Field Amplitude
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The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum is B = 510 nT. What is the amplitude of the electric field part of the wave?
Detailed Explanation
To find the electric field amplitude (E) from the magnetic field amplitude (B), we can use the relationship E = cB, where c is the speed of light in vacuum. Substituting the values will yield the electric field amplitude.
Examples & Analogies
Think of a tug-of-war game where one team trying to pull the rope (representing the electric field) is always matched by the other side's effort (the magnetic field). The strength of one must correspond to the strength of the other to maintain balance, just as E and B do in electromagnetic waves.
Exercise 8.8: Determining Wave Parameters
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Suppose that the electric field amplitude of an electromagnetic wave is E = 120 N/C and that its frequency is n = 50.0 MHz. (a) Determine, Bβ, Ο, k, and Ξ». (b) Find expressions for E and B.
Detailed Explanation
For this exercise, we calculate several parameters of the electromagnetic wave, including the magnetic field amplitude (Bβ) using Bβ = Eβ/c, angular frequency (Ο) as Ο = 2Οf, the wave number (k) as k = 2Ο/Ξ», and the wavelength (Ξ») using the relation c = fΞ». We will find the expressions for E and B using their sinusoidal expressions relating to the given frequencies and amplitudes.
Examples & Analogies
This can be likened to measuring the sound produced by musical instruments. The amplitude of the sound resonates in the air (electric field) while the vibrations create a compatible echo (magnetic field); both need to work together to create a harmonious sound.
Exercise 8.9: Photon Energy
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The terminology of different parts of the electromagnetic spectrum is given in the text. Use the formula E = hn (for energy of a quantum of radiation: photon) and obtain the photon energy in units of eV for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation?
Detailed Explanation
This exercise requires the use of the relationship E = hn to calculate the energy of photons for various electromagnetic radiation parts. Different energy levels indicate the type of radiation producedβfrom radio waves (low energy) to gamma rays (high energy)βand this energy relates directly to their production sources and interactions with matter.
Examples & Analogies
Imagine the energy from sunlight warming the earth. The varying hues (colors) of light correspond to different energy levels and sources. Just like we experience the heat from different sourcesβsome stronger (like the sun's rays) and others milder (like a flickering candle)βthe photon energies reflect how different electromagnetic waves are emitted.
Exercise 8.10: Sinusoidal Wave Analysis
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In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 Γ 10^10 Hz and amplitude 48 V mβ1.
(a) What is the wavelength of the wave?
(b) What is the amplitude of the oscillating magnetic field?
(c) Show that the average energy density of the E field equals the average energy density of the B field.
Detailed Explanation
To solve these questions, we first calculate the wavelength using Ξ» = c/f. Next, we find the magnetic field amplitude using B = E/c. The average energy density for both fields can be derived using the formulas for energy density in an electromagnetic field to show they are equal, emphasizing the energy's distribution across electric and magnetic components.
Examples & Analogies
Think of a synchronized dance where the dancers (waves) are perfectly in tune with one another. As they move in harmony (field oscillations), both sides contribute equally to the overall performance (energy distribution), ensuring a balanced and beautiful show.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Capacitance and Voltage: Understanding how to calculate capacitance and the relationship between voltage and charge in a capacitor.
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Displacement Current: Application of formulas to find the displacement current in circuits with alternating currents.
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Kirchhoff's Laws: Understanding the application of Kirchhoffβs rules to circuits with capacitors.
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Electromagnetic Wave Properties: Calculating magnetic fields generated by electromagnetic waves and understanding the relationships among electric and magnetic fields.
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The exercises serve not only as a review but also challenge the students to apply learned concepts creatively and analytically.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of calculating capacitance for a parallel plate capacitor with given dimensions.
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Example illustrating the relationship between electric and magnetic fields in an electromagnetic wave.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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If the charge is building up, the current grows, Displacement current in the circuit flows.
π Fascinating Stories
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Imagine a capacitor in a circuit filled with buzzing energy. As plates charge, an unseen current flows, linking electric and magnetic fields, creating waves.
π§ Other Memory Gems
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CIND (Capacitance, Induction, Net Current, Displacement Current) - remember these are key concepts in electromagnetism.
π― Super Acronyms
EMS (Electric Magnetic Symmetry) - helps recall the perpendicular nature of electromagnetic fields.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Capacitance
Definition:
The ability of a system to store charge per unit voltage, measured in Farads.
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Term: Displacement Current
Definition:
A term Maxwell introduced that accounts for changing electric fields contributing to magnetic fields in regions without conduction current.
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Term: Electromagnetic Wave
Definition:
A wave that consists of oscillating electric and magnetic fields perpendicular to each other, propagating through space.
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Term: Kirchhoff's Junction Rule
Definition:
A principle stating that total current entering a junction must equal total current leaving that junction.
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Term: Electric Flux
Definition:
The quantity of electric field lines passing through a surface area.