Ch 27: Current and Resistance

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 27: Current and Resistance

Problem 50

Chapter 27, Problem 50

Variations in the resistivity of blood can give valuable clues about changes in various properties of the blood. Suppose a medical device inserts microelectrodes into a 1.5-mm-diameter vein at positions 5.0 cm apart. What is the blood resistivity if a 9.0 V potential difference causes a 230 μA current through the blood in the vein?

Verified step by step guidance

Step 1: Recall the formula for resistance in terms of resistivity: \( R = \frac{\rho \cdot L}{A} \), where \( R \) is the resistance, \( \rho \) is the resistivity, \( L \) is the length of the conductor (in this case, the vein), and \( A \) is the cross-sectional area of the conductor.

Step 2: Use Ohm's Law to find the resistance \( R \): \( R = \frac{V}{I} \), where \( V \) is the potential difference (9.0 V) and \( I \) is the current (230 μA or \( 230 \times 10^{-6} \) A). Substitute the given values into this formula to calculate \( R \).

Step 3: Calculate the cross-sectional area \( A \) of the vein. The vein is cylindrical, so \( A = \pi r^2 \), where \( r \) is the radius of the vein. The diameter is given as 1.5 mm, so \( r = \frac{1.5}{2} \) mm = 0.75 mm = \( 0.75 \times 10^{-3} \) m. Substitute this value into the formula for \( A \).

Step 4: Rearrange the resistivity formula \( \rho = \frac{R \cdot A}{L} \) to solve for \( \rho \). Here, \( L \) is the distance between the electrodes, which is given as 5.0 cm = 0.05 m. Substitute the values of \( R \), \( A \), and \( L \) into this formula.

Step 5: Perform the calculations to find the resistivity \( \rho \). Ensure all units are consistent (e.g., meters for length, square meters for area, ohms for resistance) before substituting into the formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Resistivity

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current. It is defined as the resistance of a unit cube of the material and is typically measured in ohm-meters (Ω·m). In biological contexts, such as blood, resistivity can vary due to factors like temperature, composition, and cellular structure, making it a useful parameter for diagnosing health conditions.

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering and physics that relates voltage (V), current (I), and resistance (R) in a circuit. It states that V = I × R, meaning the voltage across a conductor is directly proportional to the current flowing through it, with resistance as the proportionality constant. This law is essential for calculating resistivity when the voltage and current are known.

Geometric Factors in Resistance

The resistance of a cylindrical conductor, such as a blood vessel, depends on its length and cross-sectional area. The formula for resistance (R) is R = ρ(L/A), where ρ is resistivity, L is the length of the conductor, and A is its cross-sectional area. In this case, the diameter of the vein and the distance between the electrodes are crucial for determining the effective resistance and, consequently, the resistivity of the blood.

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Textbook Question

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Textbook Question

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