A current `'I'` flows along an infinitely long straight conductor. If `'r'` is the perpendicular distance of a point from the lower end of the conductor, then the magnetic induction `B` is given by

Options

A. `1//r^(2)`
B. 1/r
C. `1//r^(3)`
D. `1//sqrt(r )`

Answer

To solve the problem of finding the magnetic induction \( B \) at a point due to an infinitely long straight conductor carrying a current \( I \), we can use the principles of magnetism, specifically the Biot-Savart Law. Here’s a step-by-step solution:

### Step 1: Understanding the Setup
We have an infinitely long straight conductor carrying a current \( I \). We want to find the magnetic field \( B \) at a point \( P \) which is at a perpendicular distance \( r \) from the lower end of the conductor.

**Hint:** Visualize the scenario with the conductor and the point \( P \) to understand the geometry involved.

### Step 2: Applying Biot-Savart Law
According to the Biot-Savart Law, the magnetic field \( B \) at a distance \( r \) from a straight conductor carrying a current \( I \) can be expressed as:
\[ B = \frac{\mu_0 I}{4\pi} \int \frac{dL \sin \theta}{r^2} \]
where \( dL \) is an infinitesimal length of the conductor, \( \theta \) is the angle between the current element and the line connecting the element to the point \( P \), and \( r \) is the distance from the current element to the point.

**Hint:** Remember that for an infinitely long straight wire, the angles \( \alpha \) and \( \beta \) will simplify the calculation.

### Step 3: Analyzing Angles
In our case:
- The angle \( \alpha \) (angle from the upper end) is \( 90^\circ \).
- The angle \( \beta \) (angle from the lower end) is \( 0^\circ \).

Thus, we have:
\[ \sin(90^\circ) = 1 \quad \text{and} \quad \sin(0^\circ) = 0 \]

**Hint:** Use the sine values to simplify the expression for \( B \).

### Step 4: Simplifying the Expression
Substituting the sine values into the Biot-Savart Law, we get:
\[ B = \frac{\mu_0 I}{4\pi} \cdot \frac{1}{r} \cdot 1 \]
This leads to:
\[ B = \frac{\mu_0 I}{4\pi r} \]

**Hint:** Focus on the relationship between \( B \) and \( r \) to identify the proportionality.

### Step 5: Conclusion
From the expression \( B = \frac{\mu_0 I}{4\pi r} \), we can see that the magnetic induction \( B \) is inversely proportional to the distance \( r \):
\[ B \propto \frac{1}{r} \]

Thus, the correct option is that the magnetic induction \( B \) is inversely proportional to \( r \).

**Final Answer:** The magnetic induction \( B \) is inversely proportional to \( r \) (Option 2).