Problem 42 - University Physics with Modern Physics

Question

Show that the quantity \(\sqrt{L} / C\) has units of resistance (ohms).

Step-by-Step Solution

Answer

The quantity \( \frac{\sqrt{L}}{C} \) has units of resistance (ohms) because its derived unit simplifies to ohms.

Step 1: Understand the Units

To prove that \( \frac{\sqrt{L}}{C} \) has units of resistance, we need to identify the units of each component. Inductance \( L \) is measured in henrys (H), and capacitance \( C \) is measured in farads (F). Resistance (R) is measured in ohms (\(\Omega\)).

Step 2: Recall Unit Equivalences

A henry (H) is equivalent to an ohm-second (\(\Omega \cdot s\)), and a farad (F) is equivalent to a second per ohm (\(\frac{s}{\Omega}\)). Therefore, we have \(H = \Omega \cdot s\) and \(F = \frac{s}{\Omega}\).

Step 3: Express \( \sqrt{L} \) in Terms of Basic Units

The unit of \( L \), which is henrys, is \( \Omega \cdot s \). Thus, \( \sqrt{L} \) has units of \( \sqrt{\Omega \cdot s} \).

Step 4: Express \( \frac{\sqrt{L}}{C} \) with Units

Now calculate the units of \( \frac{\sqrt{L}}{C} \):\[\frac{\sqrt{L}}{C} = \frac{\sqrt{\Omega \cdot s}}{s/\Omega} = \frac{\sqrt{\Omega \cdot s} \cdot \Omega}{s}\]

Step 5: Simplify the Resulting Units

Simplify the expression from Step 4:\[= \sqrt{\Omega \cdot s} \cdot \frac{\Omega}{s} = \Omega \cdot \sqrt{1} = \Omega\]Thus, \( \frac{\sqrt{L}}{C} \) has units of ohms, demonstrating that it indeed has units of resistance.

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in the study of electricity. It establishes a relationship between voltage, current, and resistance. Simply put, it states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points. The formula is expressed as:\[ V = I \cdot R \]Where:

• \( V \) is the voltage (in volts)
• \( I \) is the current (in amperes)
• \( R \) is the resistance (in ohms)

The concept is crucial because it helps us analyze electrical circuits. By using Ohm's Law, we can determine the third variable if we know any two of the parameters. Understanding this relationship also aids in the design and safe operation of electrical equipment.

Inductance

Inductance is a property of an electrical conductor that opposes changes in current. It is a measure of how efficiently a loop of wire will store energy in a magnetic field when electricity flows through it. Measured in henrys (H), inductance comes from the work done in building up the magnetic field around a coil when the electric current changes. Main points to remember about inductance:

• High inductance means a strong opposition to changes in current.
• Inductors are key components in transformers, motors, and other electrical devices.
• A higher number of coils or turns increases inductance.

Understanding inductance helps in designing circuits to control the flow of electricity and reduce undesired electronic interference.

Capacitance

Capacitance is the ability of an electronic component, or a circuit, to store and release electrical energy. It is the property of a system that allows it to store an electric charge. Capacitance is measured in farads (F). It depends on the area of the plates, the distance between the plates, and the dielectric material in between. Essential aspects of capacitance include:

• Capacitors are devices designed to hold and discharge electric charge, often used for filtering in electronic circuits.
• The larger the capacitance, the more charge can be stored.
• Capacitance is affected by the surface area of the plates and their separation distance.

Understanding capacitance is vital for designing efficient and predictable electronic systems.

Unit Conversion

Unit conversion is an essential skill, particularly in physics, as it allows for coherence and accuracy in calculations. To convert between units, we use conversion factors which act like a "bridge" between two differing units.When tackling physics problems:

• Always double-check which units your result should be in, whether metric or imperial.
• Use dimensional analysis to ensure your calculations maintain unit consistency.
• Convert complex units by breaking them down into basic components, facilitating understanding and computation. For instance, converting \( \text{H} \) to \( \Omega \cdot s \) or \( \text{F} \) to \( \frac{s}{\Omega} \).

In the context of this problem, unit conversion elucidates why \( \frac{\sqrt{L}}{C} \) results in resistance, using the equivalences \( H = \Omega \cdot s \) and \( F = \frac{s}{\Omega} \).