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Q. 42

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Physics for Scientists and Engineers: A Strategic Approach with Modern Physics
Found in: Page 871

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Short Answer

A spherical balloon with a volume of is in a uniform, vertical magnetic field. A horizontal elastic but conducting wire with resistance circles the balloon at its equator. Suddenly the balloon starts expanding at . What is the current in the wire later?

The current in wire is

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Step by Step Solution

Step: 1 Magnetic Field:

The magnetic field is uniform in this situation, even though the loop area varies as the coil tightens. The induced emf, as defined by Faraday's law, is the change in magnetic flux inside the loop, and it is provided by equation.

Where does the flux through the loop, which is the amount of magnetic field that travels across an area loop, originate from

This expression of into equation to get by

The area of the sphere is , and as the volume expands, this means the radius of the sphere changes. So, we use the expression of into equation and differentiate it for

Step: 2 Finding the value of radius:

Since the volume increases at a constant rate of, we can calculate the radius expansion rate by multiplying by

The expression of into equation to get

We ignored the significance of. . The starting volume is, and it increases to over time, allowing us to get by.

The final radius by

Step: 3 Finding the value of current:

Putting the values of and into the equation,

To get current, we utilise ohm's law.

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