IHateMayonnaise
Feb17-11, 11:58 AM
1. The problem statement, all variables and given/known data
I just took a test, and I am very unsure of the validity of how I approached the problem. Just looking for some feedback cause this is bugging me!
There is a circuit in the xz-plane (vertical), and the circuit has on it a gate. The gate is a pendulum of length L, which swings back and forth at some velocity (given), and when it is straight up and down (theta=0) it makes contact with the other lead and completes the circuit. All the time there is a CONSTANT magnetic field B pointing normal to the circuit (+y). What is the induced EMF?
The velocity as a function of time is given:
\dot{x}=\omega x_o \cos\left(\omega t\right)
2. Relevant equations
\mathcal{E}=-\frac{1}{c} \frac{d \Phi}{dt}
\Phi = \oint_S \mathbf{B}\cdot\mathbf{dS}
3. The attempt at a solution
The initial conditions can be deduced fairly easily (x_o is the horizontal amplitude of the pendulum): \dot{x}(0)=0, x(0)=x_o. From the initial conditions and integrating,
x(t) = x_o\left[\sin\left(\frac{n\pi t}{2\tau}\right)+1\right]
My (probably flawed) methodology is what follows. I thought it easier to think of the circuit as staying constant (eg no gate) and the field is pulsing at some rate, which I assume to directly correspond to that of the pendulum. From this we can then calculate the induced emf. Basically from here, all I did was take the above equation, and instead of the position amplitude x_o I substituted the field amplitude B_o.
B(t) = B_o\left[\sin\left(\frac{n\pi t}{2\tau}\right)+1\right]
From here I just took the derivative and said I was done, but I feel so, so dirty and I know I goofed this up. Thoughts? Thanks yall
IHateMayonnaise
I just took a test, and I am very unsure of the validity of how I approached the problem. Just looking for some feedback cause this is bugging me!
There is a circuit in the xz-plane (vertical), and the circuit has on it a gate. The gate is a pendulum of length L, which swings back and forth at some velocity (given), and when it is straight up and down (theta=0) it makes contact with the other lead and completes the circuit. All the time there is a CONSTANT magnetic field B pointing normal to the circuit (+y). What is the induced EMF?
The velocity as a function of time is given:
\dot{x}=\omega x_o \cos\left(\omega t\right)
2. Relevant equations
\mathcal{E}=-\frac{1}{c} \frac{d \Phi}{dt}
\Phi = \oint_S \mathbf{B}\cdot\mathbf{dS}
3. The attempt at a solution
The initial conditions can be deduced fairly easily (x_o is the horizontal amplitude of the pendulum): \dot{x}(0)=0, x(0)=x_o. From the initial conditions and integrating,
x(t) = x_o\left[\sin\left(\frac{n\pi t}{2\tau}\right)+1\right]
My (probably flawed) methodology is what follows. I thought it easier to think of the circuit as staying constant (eg no gate) and the field is pulsing at some rate, which I assume to directly correspond to that of the pendulum. From this we can then calculate the induced emf. Basically from here, all I did was take the above equation, and instead of the position amplitude x_o I substituted the field amplitude B_o.
B(t) = B_o\left[\sin\left(\frac{n\pi t}{2\tau}\right)+1\right]
From here I just took the derivative and said I was done, but I feel so, so dirty and I know I goofed this up. Thoughts? Thanks yall
IHateMayonnaise