Electromagnetism

Step 1: Relate penetrating power to the energy of a photon. The penetrating power of electromagnetic radiation, like X-rays, is directly proportional to the energy of its photons. Higher energy photons are more penetrating.
Step 2: Recall the energy-wavelength relationship for a photon. The energy ( ) of a photon is inversely proportional to its wavelength ( ): where is Planck's constant and is the speed of light.
Step 3: Determine which wavelength corresponds to the highest energy. To have the largest penetrating power, the X-ray must have the highest energy. According to the formula, the highest energy corresponds to the shortest (smallest) wavelength.
Step 4: Compare the given wavelengths. The given options are , , , and . The smallest value among these is . This wavelength will have the highest energy and therefore the largest penetrating power.

Step 1: Recall the formula for the magnetic field of a long, straight wire. Ampere's Law gives the magnetic field at a perpendicular distance from a long, straight wire carrying a current : where is the permeability of free space, T m/A.
Step 2: Identify the given values and convert them to SI units. - Current A. - Distance .
Step 3: Substitute the values into the formula and calculate .

Step 1: Relate the Poynting vector to momentum. The average Poynting vector represents the average energy flux (energy per unit area per unit time) of an electromagnetic wave. The momentum flux (momentum per unit area per unit time) is given by . This momentum flux is equal to the radiation pressure exerted by the wave when it is completely absorbed.
Step 2: Consider the case of total reflection. When the wave is totally reflected, its momentum is reversed. - The initial momentum arriving per unit area per unit time is . - The final momentum leaving per unit area per unit time is . The change in momentum per unit area per unit time is the difference between the final and initial momentum fluxes:
Step 3: Determine the pressure on the surface. The radiation pressure on the surface is the force per unit area, which is equal to the magnitude of the change in momentum per unit area per unit time transferred to the surface.

Concept: Ampère’s Circuital Law states that the line integral of the magnetic field around any closed path is equal to the permeability of free space multiplied by the total current enclosed by that path. The mathematical form of Ampère’s Law is: where

• = Magnetic field
• = Infinitesimal length element of the closed loop
• = Permeability of free space
• = Current enclosed by the loop

Step 1: Identify the result of the line integral.} The line integral of the magnetic field around a closed path equals multiplied by the enclosed current.
Step 2: Write the final value.} Thus, the value of the line integral is .

Concept: The Quality Factor (Q-factor) of an LCR circuit indicates how sharply the circuit resonates at its resonant frequency. Mathematically, or equivalently where

• = inductance
• = capacitance
• = resistance
• = resonant angular frequency

Step 1: Interpretation of Q-factor.} A high -factor means the circuit has:

• Low energy loss
• Narrow bandwidth
• Sharp resonance peak

Step 2: Bandwidth relation.} where is the bandwidth.
Step 3: Conclusion.} Therefore, the Q-factor describes the sharpness or selectivity of resonance in the circuit.

Concept: Maxwell’s equations describe the fundamental laws of electromagnetism. One of these equations states that the divergence of the magnetic field is zero.
Step 1: Interpret the equation.} The divergence of a field measures the net flow of the field out of a point. If it means magnetic field lines neither originate nor terminate at any point.
Step 2: Physical implication.} This indicates that magnetic monopoles do not exist. Magnetic field lines always form closed loops.
Step 3: Compare with other Maxwell equations.}

• Gauss’s law for electricity relates electric field divergence to charge density.
• Faraday’s law describes electromagnetic induction.
• Ampère–Maxwell law relates magnetic fields to current and changing electric fields.

Thus, the equation representing the absence of magnetic monopoles is