A coil of wire carrying a current may be used to create a magnetic field at the center of the coil. If the radius of the coil is doubled while the number of windings and the current remain the same, then how does this change the strength of the magnetic field at the center?

Here magnetic field at center does not change,  it remains constant .

Explanation:

The magnetic field at the center of the solenoid depends on only the current flowing through the wire and the number of turns of the coil around the solenoid .

  Since magnetic field at the center of the solenoid is given by the expression

  Here   is permeability constant whose value is equal to  .

 N is number of turns of the coil .

  I is current flowing through the coil .

The field lines near the center of the solenoid are approximately parallel , indicating a nearly uniform magnetic field.

0.11 m/s and 0.28 m/s should be correct

weight of the box is given as

here this is placed on the inclined of angle 42 degree

now this weight will have two components one is along the incline other is perpendicular to it

so here we need to apply a force parallel to incline opposite to its component of weight

so here the component of weight is given as

here we have

so it requires 75 n force to move it upwards