Magnetic field and induced EMF

[jaymode]

Here is the problem:
A coil containing 590 turns with radius 3.85 cm, is placed in a uniform magnetic field that varies with time according to B=( 1.20×10−2 T\s)t+( 3.40×10−5 T\s^4)t^4. The coil is connected to a resistor of resistance 690 Ohms, and its plane is perpendicular to the magnetic field. The resistance of the coil can be neglected.

A)Find the magnitude of the induced emf in the coil as a function of time.
I found that by taking the derivative of B and multiplying that by the number of turns time the area and got:
2.75*(0.012+(0.000102*(t^3))) V

which Mastering Physics tells me is right.

B)What is the current in the resistor at time t_0 = 5.00 s?
I tried to solve this by using the equation from A and plugging in 5 for t. Then I divided that by the resistance which is 690 ohms.

The answer I got was 9.8641*10^-5, however Mastering Physics tells me it wrong.

Where am I going wrong?

 

[Andrew Mason]

A)Find the magnitude of the induced emf in the coil as a function of time.
I found that by taking the derivative of B and multiplying that by the number of turns time the area and got:
2.75*(0.012+(0.000102*(t^3))) V

which Mastering Physics tells me is right.

B)What is the current in the resistor at time t_0 = 5.00 s?
I tried to solve this by using the equation from A and plugging in 5 for t. Then I divided that by the resistance which is 690 ohms.

The answer I got was 9.8641*10^-5, however Mastering Physics tells me it wrong.

Where am I going wrong?

Since the resistor is in series with the coil, the current through the resistor is the solution to this differential equation:

$$V = L\frac{dI}{dt} + IR$$

where V is the induced emf that you found in a).

So I think you need to solve this differential equation.

AM

 

[thepatientmental]

First, let's confirm that the formula you used for the induced EMF is correct. The formula for induced EMF in a coil is given by:

E = -N * (dΦ/dt)

Where N is the number of turns in the coil and dΦ/dt is the rate of change of magnetic flux through the coil. In this case, the magnetic field is changing with time, so the magnetic flux through the coil is also changing. We can calculate the magnetic flux through the coil using the formula:

Φ = B * A * cosθ

Where B is the magnetic field, A is the area of the coil, and θ is the angle between the magnetic field and the plane of the coil (which is perpendicular in this case). Plugging in the values given in the problem, we get:

Φ = (1.20×10−2 T/s)t + (3.40×10−5 T/s^4)t^4 * (590 turns) * (π*(0.0385 m)^2) * (cos 90°)

= 2.75*(0.012+(0.000102*(t^3))) t V

Therefore, the induced EMF can be calculated as:

E = -N * (dΦ/dt)

= -590 * (2.75*(0.012+(0.000102*(t^3))) V/s)

= -1.6225*(0.012+(0.000102*(t^3))) V

Now, let's move on to part B. To calculate the current in the resistor, we can use Ohm's law:

I = V/R

Where V is the voltage across the resistor and R is the resistance. In this case, the voltage across the resistor is the same as the induced EMF in the coil, which we calculated in part A. So, plugging in the values, we get:

I = (-1.6225*(0.012+(0.000102*(t^3)))) V / 690 Ω

= -2.3504*(0.012+(0.000102*(t^3))) mA

Therefore, at t = 5 s, the current in the resistor would be:

I = -2.3504*(0.012+(0.000102*(5^3))) mA

= -2.3504*(0.