Electric Current

Step 1: Determining the Radius of the Circle The length of the wire is given as: Since the wire is bent into a complete circle, the circumference of the circle is:

Step 2: Magnetic Field at the Centre of a Circular Loop The magnetic field at the center of a current-carrying circular loop is given by: where: - (permeability of free space) - A (current in the wire) - m (radius of the circle)

Step 3: Substituting Values Simplifying: Thus, the correct answer is:

We are given the peak value of the magnetic field of the electromagnetic wave as . The relationship between the electric field and magnetic field in an electromagnetic wave is given by the equation: where - is the peak value of the electric field, - is the speed of light in vacuum, , and - is the peak value of the magnetic field.

Step 1: We are provided with the value of and the speed of light .

Step 2: Substitute the given values into the equation :

Step 3: Perform the multiplication:

Step 4: Simplify the result: Thus, the peak value of the electric field is .

We are given that the de Broglie wavelength of a proton is twice that of an alpha particle, i.e., The de Broglie wavelength of a particle is related to its momentum by the formula: where is Planck's constant and is the momentum of the particle.

Step 1: Let the momentum of the proton be and the momentum of the alpha particle be . From the de Broglie relation, we have: Since , we can write: This simplifies to:

Step 2: The kinetic energy of a particle is related to its momentum by the equation: where is the mass of the particle.

Step 3: Let the masses of the proton and alpha particle be and , respectively. Then the kinetic energy of the proton and the alpha particle are given by: Substituting into the equation for , we get:

Step 4: The ratio of the kinetic energies is: Since the mass of the alpha particle is 4 times the mass of the proton (as it consists of 2 protons and 2 neutrons), we have: Thus, the ratio of the kinetic energies of the proton and the alpha particle is .

Conclusion: Therefore, the correct answer is .