Lab 223: Faraday's Law

**Daniella Brienza**

**Physics 2**

__Objective__:

To Faraday’s law of electromagnetic induction

To become familiar with the concepts of changing magnetic flux and induced current associated with Faraday’s Law of Induction

__Theoretical Background__:

Faraday’s Law of Induction states that a changing magnetic flux through a coil generates (induces) an emf in the wire -- called electromagnetic induction which explains that if you move a bar magnet through a loop of wire

When you push the bar magnet into the loop of wire (called a coil) the magnetic flux through the wire increases because of the magnetic field strength in the cross-sectional area of the coil increases.

Or when you pull the magnetic back out of the coil, the magnetic flux through the coil decreases

A small induced coil is placed between the large coils of which separation is the same as its radius

In this lab, you will use Helmholtz Coil instead of using a bar magnet to fluctuate magnetic flux through the induced coil

The pair of Helmholtz coils is connected to a power source of which function is that electrical current in the Helmholtz coils can linearly increase or decrease over time and hence magnetic field strength between them linearly increases or decreases over time

This effect results in a constant change in magnetic flux through the small coil and this it induces emf

The voltage (emf) induced in a coil by a changing magnetic field B(t) through the coil measured and compared with the predictions of Faraday’s Law

Magnetic flux Φ_{B} through a loop of wire is:

(1) Φ_{B }=

When the magnetic flux through the loop of wire is changing over time the loop opposes the changes by inducing emf (ε):

(2) ε = -

The negative sign in Equation 2 indicates that the induced emf (ε) and the change in magnetic flux have opposite signs

If the magnetic flux due to the Helmholtz coils increases over time through the small induced coil, the induced current in the small coil should have a direction to reduce the amount of magnetic flux change through it

If the loop of wire made N turns and cross-section area A is aligned normally with uniform magnetic field B then:

(3) ε = -NA()

The source of the magnetic field will be a pair of Helmholtz coils connected to a power source

A small coil is placed in the center between the Helmholtz Coils

The Helmholtz coils are connected to the signal generator that provides a time-varying source of current through the coils

A current sensor is connected in series with the Helmholtz Coils to measure the current through the Helmholtz coils

Magnetic field strength created by the Helmholtz coils can be theoretically calculated from the equation:

(4) B =

Where N_{h} is the number of turns of Helmholtz coil, R is the radius of Helmholtz and I is the current through the Helmholtz coils.

Since the magnetic field strength is linearly proportional to the current, we can rewrite equation 4 as follows:

(5) B(t) = 8.992*10^{-7} ()I(t)

As the Helmholtz coils are connected the signal generator and we will make the current through the coils linearly changing over time, the change in magnetic field strength over time is linearly proportional to the change in current over time. This gives us:

(6) = 8.992*10^{-7} ()()

Therefore we can combine equation 3 and 6 and have the following:

(7) ε = -8.992*10^{-7} (NA)()

In equation 7, N represents the number of turns of the small coil, A is the cross-sectional area of the small coil N_{h} is the number of turns of the Helmholtz coil and R is the radius of the Helmholtz coil

When all these parameters are fixed the emf induced in the small coil is linearly proportional to the change in current through the Helmholtz coils

Note that the sign of emf is opposite to that of change in current

__Procedure__:

Record the average radius (R) and the number of turns (N_{h}) of the Helmholtz coils and the area (m^2) and the number of turns (N) of the small coil in Data Table 1.

Log in to the lab computer using your UCID and password.

Connect the USB cable of the 850 Universal INterface to a USB port on the computer.

Connect the AC adapter power cord to an electrical outlet under the lab table.

Press the power push button the left front corner of the interface.

Open Lab 223 on the desktop.

Press the “tare” button on the magnetic field sensor

Go to the “Measurement” page on the screen and click “Signal Generator” on the left side of the screen. You will see the below panel of Signal Generator. Make sure that in the 850 Output 1, Triangle is chosen in Waveform, Frequency is set to 2 Hz. Amplitude is set to 5 V and Voltage Offset is set to 5 V. With this set of the signal generator, you will linearly increase and decrease the current through the Helmholtz coils over a certain period of time and hence the magnetic field strength between the coils will linearly increase and decrease. You will repeat this experiment with another frequency of 4 Hz after this.

Click “Record” on the bottom of the screen and after the measurement is finished within 2 seconds, you will see the below graphs. (Shown on raw datasheet).

Click on data summary at the left of the page to open the data summary panel and double click “Run.”

Go to “Analysis 1 Current” page and click the black triangle and select “EMF 2 Hz”. You will see the current vs. time graph. Determine dl(t)/dt in each region where current linearly increases and decreases.

Calculate emf using equation (7) with the values of dl(t)/dt which you found in step 11 and record emf Data Table2.

Go to “Analysis 2 Magnetic Field” page. Click the black triangle by the Run Select icon and select “EMF 2Hz.”

Calculate emf using equation (3) with the values of dB(t)/dt which you found in step 13 and record emf in Data Table2.

Go to “Analysis 3 #MF” page. Click on the black triangle by the Run Select icon and select “EMF 2Hz”

Repeat this experiment with a frequency of 4Hz.

Questions and Discussion:

Compare the emf which you have calculated based on dl(t)/dt and dB(t)/dt with experimentally measured emf. What is the % difference between them?

What if the small (induced) coil is positioned at 45° to the Helmholtz coils? Does the induced emf increase or decrease?

Helmholtz Coil

R (average radius) |
N |

10.06 cm |
500 |

Small (Induced) Coil

A (cross-sectional area) |
N (number of turns) |

5.29 𝛑 m^2 |
2000 |

Frequency = 2Hz |
Current increase region |
Current decrease region |

dl(t)/dt: Slope in current vs. time graph [A/sec] |
1.09 |
-0.98 |

dB(t)/dt: Slope in magnetic field strength vs. time graph [T/sec] |
0.01 |
-0.01 |

Experimentally measured emf [V] |
2 |
1.8 |

Frequency = 4Hz |
Current increase region |
Current decrease region |

dl(t)/dt: Slope in current vs. time graph [A/sec] |
1.17 |
-1.08 |

dB(t)/dt: Slope in magnetic field strength vs. time graph [T/sec] |
0.01 |
-0.01 |

Experimentally measured emf [V] |
2.034 |
-1.460 |

__Conclusion__**:**

In this lab, we were able to use Faraday’s law. We used this law to calculate the voltage induced

in a coil surrounded by Helmholtz coils. We studied this law under the effect of a changing

magnetic field. There was a decent amount of percent difference between the theoretical and

the experimental values. Through this lab it is understood that the induced voltage(emf) in a

circuit is proportional to the rate of change over time of the magnetic flux through the circuit.

Lastly, the induced current produces magnetic fields which tend to oppose the changes in

magnetic flux.

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