## Sensitivity of a galvanometer

Goal

To determine the current required for full-scale deflection of a galvanometer

Equipment

Digital multimeter
Galvanometer (from analog multimeter)
2 1-kilo-ohm (kW) resistors
6 100-kilo-ohm (kW) resistors
6-V battery pack

Summary of Method

You'll first connect a series circuit of the 2 1-kW resistors (referred to as the "load resistance") and the battery pack. Then you'll create your own voltmeter to measure the potential difference across one of the resistors. The voltmeter will be composed of a galvanometer in series with a chain of 100-kW resistors (this chain is referred to as the galvanometer's "multiplier resistance"). You'll measure the galvanometer deflection as a function of total voltmeter resistance. For comparison to the analog meter results, you'll also use the digital multimeter to measure the potential difference.

You'll use your measurements together with theoretical circuit equations to determine the current required to deflect the galvanometer full scale. You'll then check your work by changing the load resistance, calculating the multiplier resistance required for full-scale deflection, and then testing the result.

Prelab

Since detailed lab procedures are listed below, you may simply refer to these procedures in the Method section of your report.

Important: In your derivations, it's very important that you use the symbols defined in step 2 below.  It's also important that you express your work clearly.  Write in a large font, list equations one below the other, and leave extra space between them for the instructor to note corrections.  If you don't follow these guidelines, you may have to rewrite your work so that it can be checked.

1. Review HRW 28.6 and 28.7.
2. Make a diagram of the circuit described in Summary of Method above.  In addition to the components indicated, include the galvanometer with its multiplier resistance.  Denote the series load resistors as R1 and R2 (don't assume they're equal), the multiplier resistance as Rm, and the galvanometer coil resistance as Rg. R1 will be the resistor across which the voltmeter is placed. We're using fresh batteries so that we can ignore their internal resistances in comparison to the load resistances. (For simplicity, you may use the symbol Rv to represent the combined galvanometer and multiplier resistances.)
3. Using Kirchhoff's Laws (aka loop rules and junction rules), solve for the galvanometer current in terms of the given resistances and the terminal voltage of the battery. Express your result in simplest form. This equation will be used later to solve for the galvanometer current using measurements of the other quantities.
4. Rearrange your equation from 3) to obtain an equation for the multiplier resistance. This will be used later to solve for the multiplier resistance necessary to give a full-scale deflection for a given load resistance.

About recording your data, results, and calculations:  Use your lab journal as intended, which is as a journal.  Keep clear and complete labeled records of your work.  Evaluation of your work depends partly on how easily the instructor can find information and interpret what you did.

Method

Important Don't connect the circuit to the battery until the instructor has checked your circuit.  The galvanometer can easily be damaged by sending too much current through it.  If you burn out your galvanometer, your experiment is over.

About your work:  You'll work with a group of 1 or 2 other students.  You'll build the same circuit and collect the same data.  However, each student is expected to keep their own records and do their own calculations.

1. Your galvanometer has a letter (A-F) on the face of the meter.  Record this letter.
2. Use the digital multimeter to measure the resistances of all the resistors, the resistance of the galvanometer coil, and the terminal voltage of the battery pack. Record and label these values clearly, using the same symbols as defined previously.
3. Construct the circuit described previously. Use a chain of 6 100-kW resistors as the multiplier resistance.
5. If the circuit checks out, connect the battery. If the galvanometer deflects the wrong direction, reverse the battery leads. You should get a small positive deflection.
6. Construct a table with 5 columns. The first three columns will be Multiplier Resistance, Galvanometer Deflection (read on the 0-5 scale to the nearest tenth of a minor division), and Digital Voltmeter Reading. Be sure to record measured values (as opposed to nominal values). The next 2 columns will be reserved for calculations. Label these Equivalent Resistance and Galvanometer Current.
7. Reduce the multiplier resistance by one 100-kW resistor. Record the new set of measurements.
8. Repeat the previous step until the galvanometer deflection is between half- and full-scale. Don't reduce the multiplier resistance beyond this point. Otherwise you run the risk of damaging the galvanometer.
9. Disconnect the battery. Go on to the Calculations.

Important:  Keep all your components in an identifiable location in case it's necessary to retake data later.

Calculations

Significant figures are very important in this lab. Be wary of rounding errors.

Number your responses as given below.

1. What is the percentage change in the galvanometer deflection over the full range of multiplier resistances used?
2. Compare the result of 1) to the percentage change in the digital meter reading over the same range of multiplier resistances. Explain any differences qualitatively.
3. Calculate the equivalent resistance of the entire circuit (including galvanometer branch), using your data for your highest multiplier resistance. Start with a formula and show your substitutions.
4. Using one of your prelab equations and your data for the highest multiplier resistance, calculate the galvanometer current. Start with the equation, show your substitutions, and give the final result.
5. Record the values calculated in the previous two steps in your table.
6. Repeat your calculations of current and equivalent resistance for the other multiplier resistances. Don't show these calculations. Simply record them in the table.
7. In Graphical Analysis, plot a graph of Galvanometer Current vs. Scale Reading. Fit a straight line to the data. Document your graph and data table clearly. Record your name and your partners' names in a text box.  Also write in the text box the equation of fit with physics variables and numerical values of the coefficients with units.  Save the file with the name hlab7-X.ga3, where you replace X with the letter of your galvanometer.  Submit one file per lab group by emailing it to the instructor. Also email a copy of the file to every group member. This file must be submitted during the lab period unless you make other arrangements with the instructor.
8. Using the equation of fit from 7), calculate the current required for full-scale deflection of the galvanometer. We will call this the sensitivity of the meter.

Check

The purpose of this last section is to check your work.

1. Measure the battery voltage and record the value. If this value has changed significantly from the value that you found previously, be sure to use the new value in the following.
2. Ask the instructor for a new resistance value for R1 in the range of 2-10 kW. Using an equation from the prelab and the calculated galvanometer sensitivity, calculate the multiplier resistance needed for full-scale deflection of the galvanometer. Assume that all other circuit components (resistors and power supply) remain the same. Remember that the multiplier resistance doesn't include the galvanometer resistance.
3. Now replace the existing R1 in the circuit with the new one. Replace the multiplier resistance with the value calculated in the previous step. This may require finding a combination of resistors that will yield the desired value to 3 significant figures.
4. Connect the battery to the circuit and record the galvanometer deflection.
5. Calculate the percentage difference between the measured deflection and the expected value of 5.00.
6. Check with the instructor about whether your percentage difference is acceptable.

Discussion and Conclusion

Summarize what you did, how you did it, and what you found in this lab. Discuss errors that could contribute to differences between experimental and theoretical values.  Submit your lab book by the due date.