A current carrying loop is placed in a uniform magnetic field in four different orientations, I, II,

Q: Why does the loop have potential energy in a magnetic field?

Just like a mass has gravitational potential energy depending on its height, a magnetic dipole (like our current loop) has magnetic potential energy depending on its orientation relative to the external magnetic field. The field exerts a torque to align the dipole, and the energy is lowest when they are perfectly aligned.

Q: How do we determine the direction of the magnetic moment M?

The magnetic moment $\mathbf{M}$ always points in the direction of the area vector $\hat{\mathbf{n}}$. You can find this using the Right-Hand Rule: curl the fingers of your right hand in the direction of the current, and your thumb will point in the direction of $\mathbf{M}$ and $\hat{\mathbf{n}}$.

Q: Why is the energy maximum when the angle is 180 degrees?

At $180^\circ$, the dipole is pointing exactly opposite to the magnetic field. This is a state of unstable equilibrium. It's like balancing a pencil on its tip—it takes maximum energy to keep it there, and any slight nudge will cause it to flip to the lower energy state.

Q: What does a negative potential energy mean here?

Potential energy is a relative concept. Here, we define the zero-energy state as when the dipole is perpendicular to the field ($\theta = 90^\circ$). When the dipole aligns more with the field ($\theta < 90^\circ$), it drops to a lower energy state, which is mathematically represented as a negative value.

Q: Can we just look at the diagrams and guess the answer without math?

Yes, with practice! The more the normal vector points *against* the magnetic field, the higher the energy. The more it points *with* the field, the lower the energy. Case I is fully against (highest), Case IV is partially against, Case II is neutral, and Case III is partially with (lowest).

Question

A current carrying loop is placed in a uniform magnetic field in four different orientations, I, II, III and IV. Arrange them in the decreasing order of potential energy.

Options:

(A) I > III > II > IV

(B) I > II > III > IV

(C) I > IV > II > III

(D) III > IV > I > II

Sigma Percentile · Accepted Answer

### The Magnetic Dipole

Imagine a current-carrying loop placed in a uniform magnetic field. This loop isn't just a piece of wire; it behaves exactly like a tiny bar magnet, or a **magnetic dipole**.

Every magnetic dipole has a **magnetic moment**, denoted by $\mathbf{M}$. The direction of this magnetic moment is always perpendicular to the surface of the loop, pointing in the direction of the area vector $\hat{\mathbf{n}}$. You can easily find this direction using the Right-Hand Rule: curl your fingers in the direction of the current, and your thumb will point straight along $\hat{\mathbf{n}}$.

### The Energy Equation

When you place this magnetic dipole in an external magnetic field $\mathbf{B}$, it experiences a torque that tries to align it with the field. Because of this interaction, the system possesses **potential energy**.

The formula for the potential energy $U$ of a magnetic dipole is given by the dot product:

$$U = -\mathbf{M} \cdot \mathbf{B} = -MB \cos	heta$$

Here, $	heta$ is the crucial variable—it is the angle between the normal vector $\hat{\mathbf{n}}$ and the magnetic field $\mathbf{B}$. By carefully analyzing this angle in different orientations, we can determine the energy state of the loop.

### Analyzing the Orientations

Let's break down the four orientations given in the problem:

**Case I:** The normal vector $\hat{\mathbf{n}}$ is pointing exactly opposite to the magnetic field $\mathbf{B}$. This means $	heta = 180^\circ$. 
Substituting this into our formula, we get $U_1 = -MB \cos(180^\circ) = -MB(-1) = +MB$. This is the **maximum possible potential energy**, representing a state of highly unstable equilibrium.

**Case II:** The normal vector $\hat{\mathbf{n}}$ is pointing straight down, making it perfectly perpendicular to the magnetic field. Here, $	heta = 90^\circ$. 
Since $\cos(90^\circ) = 0$, the potential energy simply vanishes: $U_2 = 0$.

**Case III:** The normal vector $\hat{\mathbf{n}}$ is pointing down and to the right, creating an **acute angle** (less than $90^\circ$) with the magnetic field. 
For any acute angle, the cosine is positive. Therefore, the potential energy $U_3$ will be negative. For instance, if $	heta = 60^\circ$, $U_3 = -0.5MB$.

**Case IV:** The normal vector $\hat{\mathbf{n}}$ points up and to the left, forming an **obtuse angle** (greater than $90^\circ$) with the magnetic field. 
The cosine of an obtuse angle is negative. The two negative signs in our formula cancel out, resulting in a positive potential energy. If $	heta = 120^\circ$, $U_4 = +0.5MB$.

### The Final Verdict

Now that we have evaluated the relative energies, let's arrange them in decreasing order (from highest to lowest):

1. **Highest Energy:** Case I ($+MB$)
2. **Second Highest:** Case IV ($+0.5MB$)
3. **Neutral Energy:** Case II ($0$)
4. **Lowest Energy:** Case III ($-0.5MB$)

Mathematically, this is $U_1 > U_4 > U_2 > U_3$.

Therefore, the correct decreasing order of potential energy is **I > IV > II > III**.

Animated Solution for Physics - Magnetic Effects of Current: A current carrying loop is placed in a uniform magnetic field in four different orientations, I, II, III and IV. Arrange them in the decreasing order of potential energy.

Select Answer:

Visualized Solution

\text{Magnetic Loop Orientations}

\text{Potential Energy of a Magnetic Dipole}

\text{Case I: } \theta = 180^\circ

\text{Case II: } \theta = 90^\circ

\text{Case III: Acute Angle}

\text{Case IV: Obtuse Angle}

\text{Decreasing Order of Potential Energy}

\text{Equilibrium States}

The Sigma Insight: Magnetic Moment of Current Loop

The Magnetic Dipole

The Energy Equation

Analyzing the Orientations

The Final Verdict

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