The fundamental frequency of the G string of a guitar is f=196Hz. The fundamental vibrational mode of the string is described by the standing wavey(t,x)=acos(2πft)sin(πxL)where L=65cm is the length of the string and a=1mm is the amplitude of the oscillations. What would be the peak emf in Volts induced across the ends of the string if the guitar is played in a B=2Tmagnetic field. Assume that the magnetic field is perpendicular to the plane in which the string vibrates.
To find the peak emf induced across the ends of the string, we need to determine the maximum rate of change of the magnetic flux through the string.
The magnetic flux, Φ, through a loop is given by the equation:
Φ = ∫ B · dA,
where B is the magnetic field and dA is an elemental area vector perpendicular to the area.
In this case, the string oscillates in a plane perpendicular to the magnetic field, so the area vector will be parallel to the magnetic field. Therefore, the angle between the area vector and the magnetic field is 0 degrees.
The maximum rate of change of magnetic flux (dΦ/dt) is given by:
dΦ/dt = -B ∫ dA/dt,
where B is the magnetic field and dA/dt is the rate of change of the area. In this case, the area of the loop does not change with time, so dA/dt is zero.
Therefore, the maximum rate of change of magnetic flux is zero, and hence the induced emf is zero.
To find the peak electromotive force (emf) induced across the ends of the string when it is played in a magnetic field, we need to apply Faraday's law of electromagnetic induction.
Faraday's law states that the emf induced in a circuit is equal to the rate of change of magnetic flux through the circuit. In this case, the string acts as a circuit, and the magnetic field passes through it.
The magnetic flux (Φ) is given by the product of the magnetic field strength (B) and the area (A) that the magnetic field passes through. In this case, the area is the cross-sectional area of the string.
To calculate the induced emf, we need to determine the rate of change of magnetic flux. Since the magnetic field is perpendicular to the plane in which the string vibrates, the area is constant.
Now, let's calculate the rate of change of magnetic flux:
Φ = B * A
Since the magnetic field is perpendicular to the plane of vibration, the area is equal to the length of the string (L) multiplied by its diameter (d) which is given by twice the amplitude (a):
A = L * 2a
Next, we need to find the rate of change of magnetic flux, which is the derivative of Φ with respect to time (t):
dΦ/dt = d(B * A)/dt
Since A is constant, the derivative is only taken with respect to B:
dΦ/dt = B * dA/dt
To find dA/dt, we need to differentiate A with respect to time. The only term that changes with time is the amplitude, so:
dA/dt = 2 * da/dt
Now, we have all the elements to calculate the induced emf:
emf = dΦ/dt = B * dA/dt = B * 2 * da/dt
The derivative da/dt can be obtained by differentiating the equation for the standing wavey(t,x):
da/dt = -2πf * acos(2πft) * sin(πx/L)
Finally, we substitute the given values into the equation to find the peak emf in volts:
emf = B * 2 * da/dt = B * (-4π²f * cos(2πft) * sin(πx/L))
Please note that to get the actual value of emf, you would need to substitute the appropriate values for time (t) and position (x).