Which of the following sets have different dimensions?
(a) Pressure, Young's modulus, Stress
(b) Emf

Q: Why do stress and pressure have the same dimensions even though they are different physical quantities?

Both stress and pressure are defined as force divided by area ($F/A$). While pressure is an external force per unit area and stress is an internal restoring force per unit area, their dimensional formulas are identical: $[M L^{-1} T^{-2}]$.

Q: Is EMF a force?

No, despite its name (Electromotive Force), EMF is not a force. It is the work done per unit charge by a source (like a battery), which means it has the same dimensions as electric potential and potential difference: $[M L^2 T^{-3} A^{-1}]$.

Q: How do I quickly find the dimension of Electric Flux?

Electric flux ($\Phi_E$) is the dot product of the electric field ($E$) and area vector ($A$). Since $E = F/q$, its dimension is $[M L T^{-3} A^{-1}]$. Multiplying by area $[L^2]$ gives $[M L^3 T^{-3} A^{-1}]$.

Q: Are heat and work exactly the same thing?

Heat and work are both modes of energy transfer. While they are conceptually different (heat is driven by temperature difference, work is driven by force), they both represent energy and thus share the same dimensional formula: $[M L^2 T^{-2}]$.

Q: What is the dimension of Dipole Moment?

Electric dipole moment ($p$) is defined as the product of the magnitude of one of the charges ($q$) and the distance between them ($d$). Since $q = It$ (dimension $[A T]$) and distance has dimension $[L]$, the dimension of dipole moment is $[L T A]$.

Question

Which of the following sets have different dimensions?

Options:

(A) Pressure, Young's modulus, Stress

(B) Emf, Potential difference, Electric potential

(C) Heat, Work done, Energy

(D) Dipole moment, Electric flux, Electric field

Sigma Percentile · Accepted Answer

### Analyzing the Setup

In this problem, we are tasked with identifying the set of physical quantities that do not share the same dimensional formula.

Dimensional analysis is a powerful tool in physics that allows us to categorize quantities based on their fundamental nature.

To solve this, we must evaluate the dimensions of each quantity listed in the options by recalling their defining formulas.

### Evaluating the Options

Let us begin with the first option, which includes Pressure, Young's modulus, and Stress.

**Pressure** is defined as the external force applied per unit area, while **Stress** is the internal restoring force per unit area.

Both are mathematically represented as $\frac{F}{A}$, giving them the dimensional formula of $[M L^{-1} T^{-2}]$.

**Young's modulus** is the ratio of stress to strain. Since strain is a dimensionless quantity, Young's modulus inherits the exact same dimensions: $[M L^{-1} T^{-2}]$.

Moving on to the second option, we have Electromotive Force (EMF), Potential difference, and Electric potential.

Despite the word "force" in EMF, all three of these quantities represent the **work done per unit charge** ($\frac{W}{q}$).

Consequently, they all share the identical dimensional formula of $[M L^2 T^{-3} A^{-1}]$.

The third option lists Heat, Work done, and Energy.

This is a straightforward grouping. Heat and work are simply different modes of transferring energy.

Therefore, all three quantities are fundamentally measures of energy and possess the same dimensions: $[M L^2 T^{-2}]$.

### The Final Verdict

Finally, we examine the fourth option: Dipole moment, Electric flux, and Electric field.

The **Electric dipole moment** ($p$) is the product of charge and distance ($q 	imes d$), yielding a dimension of $[L T A]$.

The **Electric field** ($E$) is the force experienced per unit charge ($\frac{F}{q}$), which gives it a dimension of $[M L T^{-3} A^{-1}]$.

**Electric flux** ($\Phi_E$) is the product of the electric field and area ($E 	imes A$), resulting in a dimension of $[M L^3 T^{-3} A^{-1}]$.

It is abundantly clear that these three quantities have completely different defining formulas and, consequently, **different dimensions**.

Thus, this is the correct set we were looking for.

Animated Solution for Physics - Physics and Measurement: Which of the following sets have different dimensions?

Select Answer:

Visualized Solution

$Dimensional Analysis$

$Option (a)$

$Option (b)$

$Option (c)$

$Option (d)$

The Sigma Insight: Dimensional Analysis

Analyzing the Setup

Evaluating the Options

The Final Verdict

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