Chapter 20, Problem 34P

To determine

The direction of compass needle point at a given location

Answer to Problem 34P

The compass needle will point in the $49°$ North-West.

Explanation of Solution

Given:

From a straight vertical current-carrying wire horizontal compass is placed 18 cm due south.

The current in the wire is 35 A downward.

The Earth’s magnetic field $0.45\times {10}^{-4} \text{T}$ .

The magnetic declination is $0°$ that is facing towards North.

  

Formula used:

Magnetic field in the current-carrying wire is calculated as

  $B=\frac{{\mu }_{0}I}{2\pi r}$

Calculation:

Current in the wire causes the magnetic field in the West direction using right-hand rule.

  $\begin{array}{l}{B}_{wire}=\frac{{\mu }_{0}I}{2\pi r}\\ B_{wire}=\frac{{\text{(4π×10}}^{\text{-7}}\text{ T}\text{.m/A)(35 A)}}{\text{2π(0}\text{.18 m)}}\\ B_{wire}=3.89\times {10}^{-5} \text{T}\end{array}$

Thus, the compass needle will point in the direction of the net magnetic field of the system that is the inclination of the net magnetic field would be

  $\begin{array}{c}\theta ={\tan }^{-1}\frac{B_{Earth}}{B_{wire}}\\ ={\tan }^{-1}\frac{\text{0}{\text{.45×10}}^{\text{-4}}\text{T}}{\text{3}{\text{.89×10}}^{\text{-5}}\text{T}}\\ \theta =49°\text{North-West}\end{array}$

Conclusion:

The compass needle will point in the $49°$ North-West.