The formula for the EMF induced in the wire as it grows is: EMF = -πG(Do2 + 2DoGt + (Gt)2)/4
The EMF induced in the wire is given by Faraday's law of induction, which states that EMF induced in a circuit is equal to the negative rate of change of magnetic flux through the circuit. Therefore, the EMF induced in the wire can be expressed as:
EMF = -dΦ/dt
where Φ is the magnetic flux through the wire loop. Since the magnetic flux through the wire loop is proportional to the area of the wire loop, we can express this as:
EMF = -d(πD2/4)/dt
Substituting in the expression for D(t), we get:
EMF = -πG(Do + Gt)2/4
Therefore, the formula for the EMF induced in the wire as it grows is:
EMF = -πG(Do2 + 2DoGt + (Gt)2)/4
b. The resistance of the wire is given by Ohm's law as:
R = ρL/A
where ρ is the resistivity of the wire, L is the length of the wire, and A is the cross-sectional area of the wire. Since the wire is a loop, the length is equal to the circumference of the loop, and the cross-sectional area is equal to the area of the wire loop. Therefore, the resistance can be expressed as:
R = ρ(πD/To)/(πD2/4)
Substituting in the expression for D(t), we get:
R = ρTo/(π(Do + Gt)2/4)
The current in the wire is given by Ohm's law as:
I = V/R
where V is the voltage across the wire. Since the voltage across the wire is equal to the EMF induced in the wire, we can substitute the expression for the EMF in the equation above to get:
I = (πG(Do2 + 2DoGt + (Gt)2)/4)/(ρTo/(π(Do + Gt)2/4))
Therefore, the formula for the current in the wire as a function of time is:
I = (Do2 + 2DoGt + (Gt)2)/(Toρ)
c. The graph of I(t) would be a parabola that is increasing as t increases. The graph will start at I(0) = Do2/(Toρ) and will rise as t increases. The rate of increase will depend on the value of G; the higher the value of G, the faster the current increases. The graph will reach a maximum value at t = (Do/G) and will then decrease as t continues to increase. The maximum value of the current is given by I = (Do2 + 2Do2/G + (Do/G)2)/(Toρ). Beyond t = (Do/G), the graph will decrease, eventually reaching I(∞) = 0.
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