Electrostatic Potential And Capacitance

Step 1: {Formula for center of mass}

Step 2: {Substituting values}

Step 3: {Find the magnitude}
The only vector with the same magnitude is .

Step 1: {Use the nth second displacement formula}
The displacement covered in the th second is given by: where: - is the initial velocity, - is the acceleration, - is the time instant.
Step 2: {Substituting values}
Given: Step 3: {Verify the options}
Thus, the correct answer is (C) m.

{Understanding the Forces}
The electrostatic force on a charge is given by: The magnetic force on a moving charge is given by: Thus, the correct answer is (C).

The problem involves calculating the magnetic field at a point due to a current-carrying straight wire, where the ends of the wire make angles and with respect to the point . The solution can be derived using the Biot-Savart law, which gives the magnetic field generated by a current element.
Step 1: {Understanding the Biot-Savart Law}
The Biot-Savart law provides the magnetic field at a point due to a small current element . The law is given by: where: - is the permeability of free space, - is the current, - is the infinitesimal length of the wire element, - is the unit vector from the wire element to the point where the magnetic field is being calculated, - is the distance from the wire element to the point.
Step 2: {Applying the Biot-Savart Law to a Straight Wire}
For a straight current-carrying wire, the magnetic field at a point can be found by integrating the contributions from all infinitesimal elements of the wire. The result for the magnetic field due to a finite straight wire at a point is given by: where: - is the perpendicular distance from the wire to the point , - and are the angles between the line connecting the point and the ends of the wire, and the wire itself. The expression is derived by integrating the Biot-Savart law along the length of the wire. The terms and come from the geometry of the setup, which involves the angles at which the current elements contribute to the magnetic field.
Step 3: {Conclusion}
Thus, the magnetic field at point due to a straight current-carrying wire, where the ends of the wire make angles and with the point, is: Therefore, the correct option is (A).