Which of these is a correct expression for the energy density of a capacitor placed in a vacuum?

• The energy of electromagnetic radiation is directly proportional to its frequency ($E = hf$) and inversely proportional to its wavelength ($E = hc/\lambda$). Among the given options, gamma rays have the highest frequency and shortest wavelength, thus the highest energy.

The final answer is $\boxed{\text{C}}$.

• A transformer uses the principle of electromagnetic induction to change AC voltages from one level to another.

The final answer is $\boxed{\text{B}}$.

• Faraday's law, combined with Lenz's law, states that the induced electromotive force (emf) will produce a current that creates a magnetic field opposing the change in magnetic flux. This means the induced emf opposes the initial direction of the change.

The final answer is $\boxed{\text{C}}$.

Question 29: A square coil of sides $l = 0.25 \, \text{m}$ long, tightly wounded with $N = 500$ turns of wire. The coil is placed in a spatially uniform magnetic field of whose magnitude is decreasing at a rate $\frac{dB}{dt} = -0.040 \, \text{T/s}$. What is the magnitude of the induced emf in the coil?

• This phenomenon is known as mutual induction, where a changing current in one coil creates a changing magnetic field that links with a second coil, inducing an emf in the second coil.

The final answer is $\boxed{\text{E}}$.

• Magnetic flux ($\Phi_B = \int \vec{B} \cdot d\vec{A}$) is defined as the amount of magnetic field passing through a given surface. Therefore, it directly depends on the area of the surface.

The final answer is $\boxed{\text{C}}$.

• An inductor is a passive electrical component that stores energy in a magnetic field when electric current flows through it. It provides self-inductance in a circuit.

The final answer is $\boxed{\text{B}}$.

• The term "longest frequency" is typically a misnomer; it usually refers to the lowest frequency or longest wavelength. Among the given options, microwaves have the longest wavelength (and thus the lowest frequency). If it literally meant "longest frequency" as in "highest frequency", then gamma rays would be the answer. Given the common usage in physics questions, it's most likely asking for the longest wavelength/lowest frequency.

The final answer is $\boxed{\text{E}}$.

Question 35: A light bulb emits $5.00 \, \text{W}$ of power as visible light. What is the magnetic field from the light at a distance of $2.0 \, \text{m}$? (Take the speed of light as $3.00 \times 10^8 \, \text{m/s}$)

• Maxwell's four equations are:
  1. Gauss's Law for Electricity: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$
  2. Gauss's Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$
  3. Faraday's Law of Induction: $\oint \vec{E} \cdot d\vec{s} = -\frac{d\Phi_B}{dt}$
  4. Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$
• Option (B) $\oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc}$ is Ampere's Law in its original form, which is incomplete as one of Maxwell's four equations because it lacks the displacement current term ($\mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$). Option (D) is the complete Ampere-Maxwell Law. Therefore, (B) is not one of the four complete Maxwell's equations.

The final answer is $\boxed{\text{B}}$.

• The amount of magnetic field lines passing through a given surface area is the definition of magnetic flux.

The final answer is $\boxed{\text{B}}$.

• (A) An atom is made up of neutrons only: Incorrect. Atoms contain protons, neutrons, and electrons.
• (B) The charge on electrons balances the charge on protons: Correct. In a neutral atom, the number of electrons equals the number of protons, making the atom electrically neutral.
• (C) The proton rotates around the atom: Incorrect. Electrons orbit the nucleus (which contains protons and neutrons).
• (D) Number of neutrons is equal to the number of electrons: Incorrect. This is only true for specific isotopes of certain elements.
• (E) The atom rotates around the proton: Incorrect.

The final answer is $\boxed{\text{B}}$.

• When an ebonite rod is rubbed with fur, electrons (negative charges) are transferred from the fur to the ebonite rod. This gain of electrons makes the ebonite rod negatively charged.

The final answer is $\boxed{\text{D}}$.

• The electric potential ($V$) due to a point charge ($q$) at a distance ($r$) in a medium with permittivity ($\epsilon_0$) is given by the formula: $$V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$$

The final answer is $\boxed{\text{A}}$.

• Permittivity ($\epsilon_0$) has units derived from Coulomb's Law ($F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$) or capacitance ($C = \frac{\epsilon_0 A}{d}$).
  • From Coulomb's Law: $\epsilon_0 \propto \frac{q_1 q_2}{F r^2}$, so units are $\frac{\text{C}^2}{\text{N} \cdot \text{m}^2}$ or $\text{C}^2 \text{N}^{-1} \text{m}^{-2}$. (This matches III)
  • From Capacitance: $\epsilon_0 = \frac{Cd}{A}$, so units are $\frac{\text{F} \cdot \text{m}}{\text{m}^2} = \frac{\text{F}}{\text{m}}$ or $\text{F} \cdot \text{m}^{-1}$. (This matches I)
• Therefore, I and III are correct.

The final answer is $\boxed{\text{D}}$.

• This process is called electrostatic induction. When a charged object is brought near an uncharged conductor, it causes a redistribution of charges within the conductor without direct contact.

The final answer is $\boxed{\text{A}}$.

• Electric flux ($\Phi_E$) is defined as the product of the electric field component perpendicular to the surface and the area of the surface. If $\theta$ is the angle between the electric field vector ($\vec{E}$) and the normal to the surface ($\vec{A}$), then: $$\Phi_E = \vec{E} \cdot \vec{A} = EA \cos\theta$$

The final answer is $\boxed{\text{D}}$.

• This is the definition of electric potential. It represents the potential energy per unit charge at a given point in an electric field.

The final answer is $\boxed{\text{A}}$.

• The energy density ($u$) in an electric field (such as that within a capacitor) in a vacuum is given by: $$u = \frac{1}{2} \epsilon_0 E^2$$ where $\epsilon_0$ is the permittivity of free space and $E$ is the magnitude of the electric field.

The final answer is $\boxed{\text{D}}$.

• A point charge, by definition, has no physical dimension or surface. Therefore, the concept of "surface charge density" for a point charge is ill-defined. However, if the question implies the charge is uniformly distributed on the surface of a sphere with that radius, then:

• When a gold-leaf electroscope is charged, both leaves acquire the same type of charge (either positive or negative). Since like charges repel each other, the leaves diverge.

The final answer is $\boxed{\text{B}}$.