Question

A horizontal compass is placed 21 cm due south from a straight vertical wire carrying a 36 a current downward. in what direction does the compass needle point at this location? assume the horizontal component of the earth's field at this point is 0.45 â 10-4 t and the magnetic declination is 0°.

Answer 1

 
The needle of a compass will always lies along the magnetic field lines of the earth. 
A magnetic declination at a point on the earth’s surface equal to zero implies that 
the horizontal component of the earth’s magnetic field line at that specific point lies along 
the line of the north-south magnetic poles. 

The presence of a current-carrying wire creates an additional 
magnetic field that combines with the earth’s magnetic field. Since magnetic 
fields are vector quantities, therefore the magnetic field of the earth and the magnetic field of the vertical wire must be combined vectorially. 

Where:

B1 = magnetic field of the earth along the x-axis = 0.45 × 10 ⁻ ⁴ T

B2 = magnetic field due to the straight vertical wire along the y-axis

We can calculate for B2 using Amperes Law:

B2 = μ₀ i / [ 2 π R ]

B2 = [ 4π × 10 ⁻ ⁷ T • m / A ] ( 36 A ) / [ 2 π (0.21 m ) ] 
B2 = 5.97 × 10 ⁻ ⁵ T = 0.60 × 10 ⁻ ⁴ T 

The angle can be calculated using tan function:
tan θ = y / x = B₂ / B₁ = 0.60 × 10 ⁻ ⁴ T / 0.45 × 10 ⁻ ⁴ T
tan θ = 1.326

θ = 53°

The compass needle points along the direction of 53° west of north.