Question

Unit of \(\mu_{0} \mathrm{c}\) is same as that of (A) current (B) resistance (C) electric charge (D) velocity

Short answer

None of the given options (A, B, C, or D) has the same unit as the expression \(\mu_{0} \mathrm{c}\), which has the unit \(\frac{\text{T} \cdot \text{m}^{2}}{\text{A} \cdot \text{s}}\).

Step-by-step solution

Identify the components of the expression

In the expression \(\mu_{0} \mathrm{c}\), \(\mu_{0}\) is the permeability of free space value and \(\mathrm{c}\) is the speed of light.

Determine the units of each component

The permeability of free space, \(\mu_{0}\), is a constant and has a unit of \(\frac{\text{T} \cdot \text{m}}{\text{A}}\), where T is the unit for magnetic field strength (Tesla), m is the unit of length (meter), and A is the unit of electric current (Ampere). The speed of light, \(\mathrm{c}\), is also a constant and has a unit of \(\frac{\text{m}}{\text{s}}\), where m is the unit of length (meter) and s is the unit of time (second).

Determine the unit of the expression

Now, to find the unit of the expression \(\mu_{0} \mathrm{c}\), we simply need to multiply the units of each component: \[ \frac{\text{T} \cdot \text{m}}{\text{A}} \times \frac{\text{m}}{\text{s}} = \frac{\text{T} \cdot \text{m}^{2}}{\text{A} \cdot \text{s}} \]

Compare the unit of the expression with the given options

Finally, looking at the options, we check which one has the same unit as \(\frac{\text{T} \cdot \text{m}^{2}}{\text{A} \cdot \text{s}}\): (A) Current: unit is Ampere (A) (B) Resistance: unit is Ohm \(\left( \Omega = \frac{\text{V}}{\text{A}} \right)\) (C) Electric charge: unit is Coulomb (C) (D) Velocity: unit is \(\frac{\text{m}}{\text{s}}\) None of the given options has the same unit as the expression \(\mu_{0} \mathrm{c}\), which means that none of the options (A, B, C, or D) is correct.

Key concepts

Permeability of Free Space

The permeability of free space, often denoted as \(\mu_0\), is a fundamental physical constant. It's crucial in the study of electromagnetism. It represents how much resistance the vacuum presents to the formation of a magnetic field. The unit of \(\mu_0\) is \(\text{T} \cdot \text{m}/\text{A}\), where:

• \(\text{T}\) stands for Tesla, a unit of magnetic field strength.
• \(\text{m}\) is the meter, a unit of length.
• \(\text{A}\) is the Ampere, a unit of electrical current.

This constant is significant because it links electric current and magnetic fields. Understanding it helps us comprehend how electrical currents create magnetic fields in a vacuum.

Speed of Light

The speed of light, denoted as \(c\), is also a crucial constant in both physics and engineering. It reflects the speed at which light travels in a vacuum. The speed of light has a fixed value of approximately \(299,792,458\) meters per second.

• \(\text{m/s}\) is the unit. "m" for meters and "s" for seconds.

This constant is central in Einstein's theory of relativity. It means that nothing can travel faster than light in a vacuum. For many calculations and theories, \(c\) serves as a fundamental limit of speed.

Magnetic Field Strength

Magnetic field strength is measured in Tesla (\(\text{T}\)). It's a measure of the magnetizing force of a magnetic field. Tesla as a unit is named after the inventor and electrical engineer Nikola Tesla.

• 1 Tesla is equal to 1 Newton per ampere meter (\(\text{N/A} \cdot \text{m}\)).

Understanding the strength of a magnetic field helps in various applications, such as in MRI machines in medical imaging, hard drives, and even the Earth's magnetic field studies. The stronger the magnetic field, the higher the Tesla value noted.

Electrical Current

Electrical current describes the flow of electric charge and is measured in Amperes (\(\text{A}\)). It represents how many charges pass through a point in a conductor per unit of time.

• 1 Ampere is equal to 1 Coulomb of charge passing per second (\(\text{C/s}\)).

Electric current is fundamental in designing electrical circuits and systems. Whether in a simple lamp or complex computer circuit, the correct current ensures smooth and efficient operation. High currents can lead to overheating, so it's crucial to measure and control them carefully.

Dimensional Analysis

Dimensional analysis is a tool used in physics and engineering to understand the relations between different physical quantities by identifying their fundamental units.

• Helps verify formulas by checking the consistency of units.
• Aids in converting units from one system to another.

Through dimensional analysis, you can determine whether a physical equation or formula is set up correctly by ensuring that the units on both sides match. This technique simplifies understanding of the physical world by breaking complex equations into simpler unit checks.