PHYS 1434 The Wheatstone Bridge Method

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CUNY New York City College of Technology *

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1434

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Electrical Engineering

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Apr 3, 2024

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NEW YORK CITY COLLEGE OF TECHNOLOGY LAB REPORT #6 PHYS 1434 D785 Laboratory Report #6 – The Wheatstone Bridge Method Due: March 18, 2024 Group Members: Jesus G. Davina A. Lesly G.

PHYS 1434 i Laboratory No. 6 The Wheatstone Bridge Method TABLE OF CONTENTS OBJECTIVE.. ………………………… . ………………………………………………..…… ..... 1 THEORTICAL BACKGROUND.. ……………… ... ……………………………………………………..…… ..... 1 EQUIPMENT.. …………………………………………………………………………..…… ..... 2 PROCEDURE …………………………………………………………………………..… . … ..... 2 SUMMARY OF RESULTS … . ……………………………………………………..……..… ..... 3 ATTACHMENT … . ……………………………………………………………………..…… ..... 5 SAMPLE CALCULATIONS ………..……………………………………… . ……………………..……… 6 DISCUSSION AND CONCLUSION……………………………………………………………………………… ...... 6 CITATIONS……………………………………………………………………………..… .. … ...8 ATTACHMENTS ATTACHMENT A ………………………………………………………………. Data Sheet

PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method OBJECTIVE: The objective of the laboratory experiment "The Wheatstone Bridge Method" is to provide students with hands-on experience and understanding of the Wheatstone bridge technique for measuring electrical resistance, aiming to familiarize them with the principles and operation of the Wheatstone bridge method for accurate resistance measurement. Additionally, students will determine the resistivity of a metal conductor using the Wheatstone bridge setup and appropriate calculations, while also investigating and studying the relationship between the resistance of a wire, its length, and its cross-sectional area. This exploration will deepen their comprehension of electrical circuits, resistance measurements, and the fundamental principles governing electrical conductivity in materials. THEORITCAL BACKGROUND: A simple resistor has The Wheatstone bridge method is a fundamental technique used in electrical measurements, particularly for accurately determining resistance values. It relies on the principle of balancing two branches of a circuit to nullify the current flowing through a galvanometer, thereby indicating equilibrium and allowing precise resistance calculations. The key components of a Wheatstone bridge setup include four resistors arranged in a diamond shape, with a voltage source connected across one diagonal and a galvanometer across the other. When the bridge is balanced, the ratio of resistances in one branch equals the ratio in the opposite branch, leading to zero current through the galvanometer. In this laboratory experiment, students will delve into the theoretical underpinnings of the Wheatstone bridge method. They will explore the concept of resistivity, which is a material property indicating how strongly a substance opposes the flow of electric current. By measuring resistance and dimensions (length and cross-sectional area) of a metal conductor, students will calculate its resistivity using the formula ρ = R × (A / L), where ρ is resistivity, R is resistance, A is cross-sectional area, and L is length. This process not only reinforces understanding of resistance and resistivity but also illustrates the direct proportionality of resistance to length and the inverse proportionality to cross- sectional area, as described by the equation R = ρ × (L / A). Moreover, the laboratory session will involve practical applications of the Wheatstone bridge method, such as determining unknown resistances or verifying known resistances. Through hands-on experimentation and theoretical discussions, students will gain a comprehensive grasp of electrical measurements, circuit balancing techniques, and the fundamental relationships governing resistance in electrical conductors.

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PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method EQUIPMENT: • Slide-wire Wheatstone bridge • Low voltage DC power supply • Two set of coils for unknown resistance • Resistance Box • Galvanometer • Connecting wires • LCR Meter PROCEDURE: 1. Connect the slight wire Wheatstone bridge circuit as shown in figure 7.2 try to lay out all components specifically as they appear in the diagram for you. It is easier this way to verify the connections. For or use a standard resistance box, and for RX one of the five wire spools. The material, length, and diameter of the spool wire and suggested value for the resistant box for each spool are listed below. 2. Use this phone number one as an unknown resistance, set that suggestive value of the resistance R and move sliding contact B to the middle of the bridge. Using calculus it can be shown that it is best to operate the bridge in such a way that the balance is in the middle of the slide fire. The contact should be moved only when its edge is not touching the water period now gently tap the contact B. If the galvanometer shows a deflection, very the value of the resistance R of the resistance box by, say, 0.1 ohms through 0.2 ohms enter the smallest inflection possible is obtained period now adjust the position of the slide contact be near the middle of the wire with internal deflection occurs when the contact B is tapped when the final adjustment is made for girly flexion, record the value of the standard resistance R in two lengths L1 and L2. 3. Repeat Step 2 for the four other spools. Tabulate all your results for R, L1 and L2. 4. Use a LCDR meter and measure the resistance of each pool. Tabulate these along with the bridge measurements.

PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method SUMMARY OF RESULTS: Spool Number Length of wire, m Known resistance R, Ω Length L 1 , m Length L 2 , m Spool Resistance R, Ω Spool Resistance with multimeter, R x, Ω % difference 1 10.0 0.6 58 42 0.43 0.8 60 % 2 10.0 2.4 53 47 3.418 3.7 7.92 % 3 20.0 1.0 53 47 0.886 1.3 37.88 % 4 20.0 5.0 53 47 5.638 5.1 10 % 5 10.0 10.0 53 47 11.276 9.6 16 % TABLE 1 – Measurements for the Resistance Spool Number Material Resistivity, ρ , Ω m % error Experimental Average Standard 1 Copper (Cu) 1.4 x 10 -8 1.85 x 10 -8 1.77 x 10 -8 0.04 % 2 Copper (Cu) 2.5 x 10 -8 3 Copper (Cu) 1.4 x 10 -8 4 Copper (Cu) 2.1 x 10 -8 5 Alloy (Cu+Ni) 3.5 x 10 -7 33 x10 -8 0.06 % TABLE 2 – Resistivity of Materials

PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method Ratio of resistances 𝑹𝒙, 𝐬???𝐥 ? 𝑹𝒙, 𝐬???𝐥 ? 0.615 Ratio of resistances 𝑹𝒙, 𝐬???𝐥 ? 𝑹𝒙, 𝐬???𝐥 ? 0.725 Ratio of lengths 𝑳, 𝐬???𝐥 ? 𝑳, 𝐬???𝐥 ? 0.5 Ratio of lengths 𝑳, 𝐬???𝐥 ? 𝑳, 𝐬???𝐥 ? 0.5 % difference for ratios 20 % % difference for ratios 37 % TABLE 3 Ratio of resistances 𝑹𝒙, 𝐬???𝐥 ? 𝑹𝒙, 𝐬???𝐥 ? 0.216 Ratio of resistances 𝑹𝒙, 𝐬???𝐥 ? 𝑹𝒙, 𝐬???𝐥 ? 0.255 Ratio of lengths 𝑨, 𝐬???𝐥 ? 𝑨, 𝐬???𝐥 ? 0.23 Ratio of lengths 𝑨, 𝐬???𝐥 ? 𝑨, 𝐬???𝐥 ? 0.23 % difference for ratios 6.3 % % difference for ratios 10 % TABLE 4

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PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method ATTACHMENTS: Attachment A - Data Sheet

PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method SAMPLE CALCULATIONS: CONCLUSION AND QUESTIONS: In conclusion, this laboratory experiment provided us with valuable insights and practical experience in using the Wheatstone bridge method for measuring resistance accurately. By performing measurements and calculations, we were able to determine the resistivity of a metal conductor, showcasing the relationship between resistance, length, and cross-sectional area as described by fundamental physics principles. Through this hands-on approach, we solidified our understanding of electrical circuits, resistance measurements, and the fundamental phenomenon that the resistance of a wire varies directly with its length and inversely with its cross-sectional area. This experiment not only enhanced our knowledge of electrical conductivity but also reinforced the importance of precise measurement techniques in scientific investigations. As for the % difference and % error from all four tables come be seen as human error, perhaps a misreading or miscalculation.

PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method 1. Derive equation (8). Step 1: Identify Circuit Configuration → R1, R2, R3, Rx Step 2: Apply Kirchhoff's Laws → Assume the bridge is balanced. This means the voltage drop across R1 is the same as across Rx, and the voltage drop across R2 is the same as across R3. Step 3: Balance Conditions or Ratio of Dividers → Since the bridge is balanced, the ratios of the voltages are equal. So, 𝑉𝑅1 𝑉𝑅2 = 𝑉𝑅𝑥 𝑉𝑅3 Step 4: Solve for the Unknown → Cross multiplying gives us R1R3=RxR2. Solving for Rx gives us Rx = 𝑅2𝑅3 𝑅1 2. Why is the Wheatstone bridge method of measurement of the resistance called a “null method”? The bridge is a means of determining the value of an unknown resistance by making it part of the two voltage dividers in the bridge. A precision variable resistor is in the corresponding position in the other divider. The value is found by adjusting the variable resistor until the voltage difference between the two divider outputs is zero. This is indicated by a null (0) reading on a galvanometer connected to the bridge. 3. In the experiment we neglect the resistance of contacting leads used to hook up the circuit. Explain, could these affect your results? Yes, because the wires themselves have a resistance but since the wire it very short the resistance can be neglected. 4. In Fig. 7.1, the Wheatstone bridge is balanced when R 1 =5.5 ohms, R 2 =11.0 ohms, and R 3 =4.0 ohms. Find the unknown resistance R x . - R x = ( R 2 / R 1 ) * R3 Rx = ( 11.0 Ω / 5.5 Ω) * 4.0 Ω Rx = 8 Ω 5. What length of a 22 gauge American Wire Gauge (refer to Appendix) copper wire is needed to make a resistance of 500 ohms? 1. Resistivity of copper ( ρ ): 1.68 × 10 −8 ohm-meter 2. Radius of the wire ( r ): 0.6426 ×10 -3 m / 2= 0.3213 × 10 -3

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PHYS 1434 Laboratory No. 6 The Wheatstone Bridge Method 3. Cross-sectional area ( A ): π (0.0003213) 2 = 0.3243 x 10 -6 4. Resistance ( R ): 500 ohms 𝐋 = 500 (0.3243 ∗ 10 −6 ) 1.68 ∗ 10 −8 𝐋 = 9651.78 meters WORKS CITED: Kezerashvili, Roman Ya. Laboratory Experiments in College Physics: Electricity, Magnetism, Optics, Modern Physics . Gurami Pub., 2018. “Free Textbooks Online with No Catch.” OpenStax, www.openstax.org/details/college -physics. Accessed 18 Feb. 2024. .