Animated Solution for Physics - Physics and Measurement: Give the MKS units for each of the following quantities. (a) Young's modulus (b) Magnetic induction (c) Power of a lens

When faced with finding the units of various physical quantities, the most robust strategy is to trace them back to their fundamental defining equations. Let's break down each of the three quantities requested in the problem.

The Measure of Elasticity

Imagine you are standing in a lab, stretching a metallic wire. When you apply a force $F$ to a wire of cross-sectional area $A$ and original length $L$, it elongates by a small amount $\Delta L$.

Young's modulus, denoted by $Y$, is the fundamental property that tells us how stiff the material is. It is defined as the ratio of longitudinal stress to longitudinal strain:

To find the MKS unit, we look at the components. Force is measured in Newtons ($\text{N}$) and Area in square meters (${\text{m}}^2$), making the unit of stress ${\text{N/m}}^2$. Strain, however, is the ratio of two lengths ($\Delta L/L$), which means the meters cancel out, leaving it completely dimensionless!

Therefore, the MKS unit of Young's Modulus is simply inherited from the stress, which is ${\text{N/m}}^2$.

The Strength of the Field

Next, let's dive into electromagnetism. Magnetic induction, represented by the vector $\vec{B}$, describes the strength of a magnetic field.

To find its unit, we recall the magnetic force experienced by a straight current-carrying wire placed in a uniform magnetic field. The force $F$ is given by:

If we rearrange this master equation to solve for $B$, we get:

In the MKS system, Force is measured in Newtons ($\text{N}$), Current $I$ in Amperes ($\text{A}$), and Length $L$ in meters ($\text{m}$). The trigonometric term $\sin \theta$ is just a number and has no units. Plugging these in, we get the unit of $B$ as $\text{N}/(\text{A}\cdot \text{m})$.

In honor of the legendary physicist Nikola Tesla, this combination is officially named the Tesla ($\text{T}$).

Bending the Light

Finally, we step into the world of optics. The power of a lens, $P$, quantifies its ability to converge or diverge light rays. A highly curved lens bends light sharply and has a short focal length, meaning it has high power.

Mathematically, power is defined as the reciprocal of the focal length $f$:

In the MKS system, the standard unit of length is the meter ($\text{m}$). Therefore, the unit of power is simply the reciprocal of a meter, or ${\text{m}}^{-1}$.

Optometrists and physicists commonly refer to this unit as the Dioptre ($\text{D}$), but fundamentally, it remains ${\text{m}}^{-1}$.

By anchoring ourselves to the core physical formulas, we effortlessly derived the MKS units. Young's modulus is ${\text{N/m}}^2$, Magnetic induction is Tesla, and the Power of a lens is ${\text{m}}^{-1}$.