 #1
Cactus
 15
 1
 Homework Statement:

Hey, so I've been given a question to find the induced current in a conducting square within a solenoid. The solenoid has a current running that is increasing so that the magnetic flux is increasing. The question first asked for a ring inside and outside the solenoid (encompassing it), for which I derived the two formulas in red on the graph (Where initially when the ring is inside the solenoid, the current induced constantly increases, however when the radius of the ring exceeds that of the solenoid it decreases at a rapid rate (as expected). However, when I apply the same working to the square, the plotted graph does not look right
These are the details and questions:
An ideal solenoid, of radius R and n turns per unit length, has a current flowing through it. The current, I, varies with time, t, according to I=I0+at where I0 and a are constants. A conducting ring of radius, r, is placed inside the solenoid with its axis coinciding with the axis of the solenoid. The ring has a resistance per unit length of H(in units of Ω/m).
Make a plot of the current in the ring as a function of the radius of the ring. Include curves for r < R and r > R. Explain what the plot shows. (Done Below)
The ring is replaced by a conducting wire in the shape of a square. The side length of the square is l, and it is made of the same conducting material as the ring. The square is tilted by an angle, θ (the angle between the area vector for the solenoid and the area vector for the square is θ). Repeat your calculations in Part A to determine the current in the square. Include calculations for when the square is completely inside the solenoid and completely outside the solenoid only
I am just wondering firstly if when both the ring and square encompass the solenoid, the correct area to use for the flux is that of the solenoids area as both r and l are greater than R. Likewise, are the following two formulas correct for the square, or should I include a cosθ for the outside (although the angle should not matter as all flux occurs in the solenoids cross sectional area, and even if I do include a cosθ the graphs still do not connect).
 Relevant Equations:

Current = emf/resistance
Solenoid Field = u*n*I (n = no. turns per unit length)
Flux = A*B*cosθ
Current in Solenoid = I0 + at (a is some constant)
Resistance = H * length or circumference
Plot for the ring ^
Calculations for the Square ^
Plot for square without cosg on the outside calc ^
Plot for square with cosg on the outside calc ^
As can be seen the formulas for the square conductor do not connect at R, which I'm not sure if they should or if they should not as in this case the square does not fit the shape of the solenoid so the crossover occurs later (at the intersection of the two graphs) (whilst the ring r can equal R as they are both circles). Or otherwise, I am not sure if my formula derived is correct