A paramagnetic substance in the form of a cube with sides 1 cm has a magnetic dipole moment of 20 x

Q: Why do we need to convert the side of the cube from cm to m?

In physics, formulas are designed to work seamlessly with standard SI units. The magnetic moment is given in J/T (which is equivalent to A·m²) and magnetic intensity in A/m. If you leave the volume in cm³, your final answer will be off by several orders of magnitude.

Q: What is the difference between Magnetic Dipole Moment ($M$) and Intensity of Magnetisation ($I$)?

Magnetic Dipole Moment ($M$) is an extensive property; it tells you the total magnetic strength of the entire object. Intensity of Magnetisation ($I$) is an intensive property; it is the magnetic moment per unit volume ($I = M/V$). It tells you how strongly the material itself is magnetized, regardless of how big the object is.

Q: Why does the $10^{-6}$ cancel out so perfectly?

That's the beauty of well-designed physics problems! The volume was $10^{-6}$ m³ and the magnetic moment was $20 \times 10^{-6}$ J/T. When you divide them to find $I$, the $10^{-6}$ terms cancel, leaving a clean integer of 20 A/m. This is a good sign that you are on the right track.

Q: Does magnetic susceptibility ($\chi$) have units?

In the SI system, magnetic susceptibility is a dimensionless quantity. This is because it is the ratio of Intensity of Magnetisation ($I$, measured in A/m) to Magnetic Intensity ($H$, also measured in A/m). The units cancel out completely.

Q: What does a positive susceptibility value mean physically?

A positive susceptibility (like the $3.3 \times 10^{-4}$ we found) indicates that the material is paramagnetic (or ferromagnetic if it's much larger). It means the internal magnetization aligns in the same direction as the applied external magnetic field.

Question

A paramagnetic substance in the form of a cube with sides $1 	ext{ cm}$ has a magnetic dipole moment of $20 	imes 10^{-6} 	ext{ J/T}$ when a magnetic intensity of $60 	imes 10^3 	ext{ A/m}$ is applied. Its magnetic susceptibility is

Options:

(A) $3.3 	imes 10^{-4}$

(B) $3.3 	imes 10^{-2}$

(C) $4.3 	imes 10^{-2}$

(D) $2.3 	imes 10^{-2}$

Sigma Percentile · Accepted Answer

### Unlocking the Magnetic Secrets of a Paramagnetic Cube

Imagine holding a tiny cube made of a paramagnetic material. It looks ordinary, but when exposed to an external magnetic field, it awakens. The atoms inside align, and the cube develops its own magnetic identity. In this problem, we are tasked with finding the **magnetic susceptibility** of such a cube, a fundamental property that tells us exactly how "responsive" this material is to an external magnetic push.

Let's break down the physics and the math step-by-step.

### Step 1: Finding the Physical Volume

Before we can dive into the magnetic properties, we need to establish the physical space this cube occupies. The problem states that the side of the cube is $a = 1 	ext{ cm}$.

In physics, consistency in units is paramount. We must convert this length into the standard SI unit, meters:

$$a = 10^{-2} 	ext{ m}$$

The volume $V$ of a cube is simply the cube of its side length:

$$V = a^3 = (10^{-2} 	ext{ m})^3 = 10^{-6} 	ext{ m}^3$$

This tiny volume is the domain where all the magnetic magic happens.

### Step 2: The Intensity of Magnetisation ($I$)

When the external magnetic intensity $H = 60 	imes 10^3 	ext{ A/m}$ is applied, the cube develops a total magnetic dipole moment $M = 20 	imes 10^{-6} 	ext{ J/T}$.

However, total magnetic moment isn't a fair measure of the material's intrinsic property because it depends on the size of the object. A larger cube would naturally have a larger total moment. To standardize this, we define the **Intensity of Magnetisation ($I$)**. Think of it as the "density of magnetism"—the magnetic moment developed per unit volume.

$$I = \frac{M}{V}$$

Substituting our known values:

$$I = \frac{20 	imes 10^{-6}}{10^{-6}}$$

Notice how beautifully the $10^{-6}$ terms cancel out! We are left with a clean, simple integer:

$$I = 20 	ext{ A/m}$$

### Step 3: Calculating Magnetic Susceptibility ($\chi$)

Now we reach the core of the problem. **Magnetic susceptibility ($\chi$)** is the ultimate measure of a material's magnetic responsiveness. It answers the question: *For a given external magnetic push ($H$), how much magnetization ($I$) does the material actually develop?*

Mathematically, it is the ratio of the induced magnetization to the applied magnetic intensity:

$$\chi = \frac{I}{H}$$

Let's plug in our calculated $I$ and the given $H$:

$$\chi = \frac{20}{60 	imes 10^3}$$

Simplifying the fraction $\frac{20}{60}$ gives us $\frac{1}{3}$. Bringing the $10^3$ from the denominator to the numerator changes its sign to $10^{-3}$:

$$\chi = \frac{1}{3} 	imes 10^{-3}$$

Since $\frac{1}{3} \approx 0.333$, we can write:

$$\chi \approx 0.333 	imes 10^{-3}$$

To match the standard scientific notation format of our options, we shift the decimal point one place to the right, which decreases the exponent by one:

$$\chi = 3.3 	imes 10^{-4}$$

### The Final Verdict

Our calculated magnetic susceptibility is $3.3 	imes 10^{-4}$. Looking at the given choices, this perfectly matches **Option (a)**.

This problem is a fantastic reminder of how macroscopic geometric properties (like the volume of a cube) are intimately tied to the microscopic magnetic behavior of materials. Always remember to keep your units in check, and the physics will naturally unfold!

Animated Solution for Physics - Magnetic Effects of Current: A paramagnetic substance in the form of a cube with sides $1 cm$ has a magnetic dipole moment of $20 \times 1 0^{- 6} J/T$ when a magnetic intensity of $60 \times 1 0^{3} A/m$ is applied. Its magnetic susceptibility is

Select Answer:

Visualized Solution

\text{Visualizing the Paramagnetic Cube}

\text{Volume of the Cube}

\text{Calculating the Volume}

\text{Intensity of Magnetisation } (I)

\text{Substituting Values for } I

\text{Calculating } I

\text{Magnetic Susceptibility } (\chi)

\text{Substituting Values for } \chi

\text{Final Calculation}

\text{Conclusion}

The Sigma Insight: Magnetic Materials

Unlocking the Magnetic Secrets of a Paramagnetic Cube

Step 1

Step 2

Step 3

The Final Verdict

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