In a uniform magnetic field, can an Amperian loop ever have a non-zero value of its line integral with the magnetic field?

a)yes
b)No, it will always be zero
c)depends on the orientation and shape of the loop.

The correct answer is b) No, it will always be zero.

The line integral of a magnetic field over a closed loop, known as the magnetic flux, is given by Faraday's Law of electromagnetic induction. According to Faraday's Law, the magnetic flux through any closed loop is always zero in a uniform magnetic field.

To understand why this is the case, we need to consider the properties of a uniform magnetic field and the concept of magnetic field lines. In a uniform magnetic field, the field lines are parallel and evenly spaced.

Now, let's consider an Amperian loop, which is a closed loop used to calculate the magnetic field. Since the magnetic field is uniform, the magnetic field lines passing through the loop are also parallel and evenly spaced.

When we calculate the line integral of the magnetic field along the loop, we essentially sum up the product of the magnetic field strength and the differential length element along the loop. However, due to the parallel arrangement of the field lines, the dot product of the magnetic field and the differential length element is always zero.

Therefore, the line integral of the magnetic field over any closed loop in a uniform magnetic field will always be zero. In other words, the magnetic flux through the loop is always zero, regardless of the orientation or shape of the loop.

Hence, the correct answer is b) No, it will always be zero.