Question

Choose the correct statement

• A. The magnetic dipole moment is the sum of current and area of the loop, its direction is normal to the loop
• B. The magnetic dipole moment is the product of current and area of the loop, it has no direction
• C. The magnetic dipole moment is the product of current and area of the loop, its direction is normal to the loop
• D. The magnetic dipole moment is the sum of current and volume of the loop, its direction is normal to the loop

Hint:

The SI unit for magnetic dipole moment can be read directly from its product definition formula: \[ \text{Unit of } \vec{m} = (\text{Current } I) \times (\text{Area } A) = \text{Ampere} \cdot \text{meter}^2 \quad (\text{A}\cdot\text{m}^2) \] Another equivalent unit often encountered in thermodynamics and atomic physics is Joules per Tesla ($\text{J/T}$).

Answer 1

The Correct Option is C

Concept: A planar loop of wire carrying an electrical current behaves as a source of a magnetic field, exhibiting a clear magnetic field profile that resembles a traditional bar magnet with distinct north and south poles. The fundamental vector parameter used to quantify the strength and orientation of this magnetic source is the Magnetic Dipole Moment (denoted by $\vec{m}$). Let us examine a closed, flat planar loop enclosing a surface area $A$ and carrying a steady macroscopic current $I$.

Step 1: Establishing the Mathematical Definition The magnitude of the magnetic dipole moment $m$ for a single-turn current loop is directly proportional to both the magnitude of the circulating current and the geometric surface area enclosed by that path: \[ m = I \cdot A \] If the loop consists of $N$ identical, closely wound turns of wire, the total magnitude multiplies proportionally: $m = N I A$.

Step 2: Analyzing the Vector Orientation The magnetic dipole moment is a true vector quantity ($\vec{m}$). Its direction is oriented perpendicular (normal) to the flat plane of the loop. The vector equation is given by: \[ \vec{m} = I \cdot \vec{A} = I \cdot A \;\hat{n} \] Where $\hat{n}$ represents the unit vector normal to the surface of the loop. The formal direction of the normal vector $\hat{n}$ is uniquely determined using the Right-Hand Rule: Curling the fingers of your right hand along the direction of the conventional current flow around the loop causes your thumb to point directly in the vector direction of the magnetic dipole moment $\vec{m}$. Step-by-step Review of the Statements:
• Statement 1: Incorrectly states that it is the "sum" of current and area.
• Statement 2: Incorrectly states that it "has no direction," which contradicts its vector nature.
• Statement 3: Correctly states that it is the product of the current and the area of the loop, and its direction is normal to the loop plane.
• Statement 4: Incorrectly substitutes "volume" and uses the word "sum". Thus, Statement (3) is the only accurate description.