In Fig `29-40` point `P_(2)` is at perpendicular distance `R=25.1` cm from one end of a straight wire of length `L=13.6` cm carrying current `i=0.500` A. (Note that the wire is not long ) (a) What is the magnitude of the magnetic field at `P_(2)` ? (b) If the point of measurement is moved from `P_(2)` to `P_(1)` does the field magnitude increase, decrease, or remain the same ?

Answer

To solve the problem, we will use the formula for the magnetic field \( B \) at a point due to a finite straight wire carrying current. The formula is given by:

\[ B = \frac{\mu_0 I}{4\pi r} (\sin \alpha + \sin \beta) \]

where:
- \( \mu_0 \) is the permeability of free space (\( 4\pi \times 10^{-7} \, \text{T m/A} \)),
- \( I \) is the current in the wire,
- \( r \) is the perpendicular distance from the wire to the point of interest,
- \( \alpha \) is the angle subtended by the wire at the point of interest,
- \( \beta \) is the angle subtended by the wire at the other end (which is zero in this case).

### Step-by-Step Solution:

1. **Identify Given Values**:
- Length of the wire, \( L = 13.6 \, \text{cm} = 0.136 \, \text{m} \)
- Current, \( I = 0.500 \, \text{A} \)
- Perpendicular distance, \( R = 25.1 \, \text{cm} = 0.251 \, \text{m} \)

2. **Calculate Angles**:
- The angle \( \alpha \) can be found using the tangent function:
\[ \tan \alpha = \frac{L}{R} = \frac{0.136}{0.251} \]
- Calculate \( \alpha \):
\[ \alpha = \tan^{-1}\left(\frac{0.136}{0.251}\right) \]

3. **Calculate \( \sin \alpha \)**:
- Using the right triangle formed, we can find \( \sin \alpha \):
\[ \sin \alpha = \frac{L}{\sqrt{L^2 + R^2}} = \frac{0.136}{\sqrt{(0.136)^2 + (0.251)^2}} \]

4. **Substitute Values into the Magnetic Field Formula**:
- Since \( \beta = 0 \), \( \sin \beta = 0 \):
\[ B = \frac{\mu_0 I}{4\pi R} (\sin \alpha + 0) \]
- Substitute \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \), \( I = 0.500 \, \text{A} \), and \( R = 0.251 \, \text{m} \):
\[ B = \frac{(4\pi \times 10^{-7}) \times 0.500}{4\pi \times 0.251} \sin \alpha \]

5. **Calculate the Magnetic Field \( B \)**:
- After calculating \( \sin \alpha \) and substituting it back into the equation, compute \( B \):
\[ B \approx 94.9 \times 10^{-9} \, \text{T} = 94.9 \, \text{nT} \]

### Part (b): Change in Magnetic Field Magnitude
- If the point of measurement is moved from \( P_2 \) to \( P_1 \), which is closer to the wire, the magnetic field magnitude will **increase**. This is because the magnetic field strength is inversely proportional to the distance from the wire.