A measurement of the horizatal component `B__(H)` of the Earth's field at the location of Tuscon, Arizona, gave a value of `26muT` . By suspending a small magent like a compass that is free to swing in a vertical plante ,it is possible to measure the angle between the field direction and the horizontal plane, called the inclination or the dip angle `phi` .The dip angle at Tucson was measured to be `60^(@)` . Find teh magnitude at taht location .

Options

A. 52 u T
B. 26sqrt(3)uT`
C. 26sqrt(2)muT`
D. 26 uT

Answer

To find the magnitude of the Earth's magnetic field at Tucson, Arizona, we can use the relationship between the horizontal component of the magnetic field, the vertical component, and the total magnetic field. Here's a step-by-step solution:

### Step 1: Understand the components of the magnetic field
The Earth's magnetic field can be broken down into two components:
- The horizontal component \( B_H \)
- The vertical component \( B_V \)

The total magnetic field \( B \) can be calculated using the Pythagorean theorem:
\[ B = \sqrt{B_H^2 + B_V^2} \]

### Step 2: Identify the given values
From the problem statement, we have:
- The horizontal component \( B_H = 26 \, \mu T \)
- The dip angle \( \phi = 60^\circ \)

### Step 3: Relate the vertical component to the dip angle
The vertical component \( B_V \) can be expressed in terms of the total magnetic field \( B \) and the dip angle \( \phi \):
\[ B_V = B \sin(\phi) \]

### Step 4: Relate the horizontal component to the dip angle
The horizontal component can also be expressed as:
\[ B_H = B \cos(\phi) \]

### Step 5: Substitute the known values
We can rearrange the equation for \( B_H \):
\[ B = \frac{B_H}{\cos(\phi)} \]
Substituting the values:
\[ B = \frac{26 \, \mu T}{\cos(60^\circ)} \]

### Step 6: Calculate \( \cos(60^\circ) \)
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \]

### Step 7: Substitute \( \cos(60^\circ) \) into the equation
\[ B = \frac{26 \, \mu T}{\frac{1}{2}} = 26 \, \mu T \times 2 = 52 \, \mu T \]

### Step 8: Conclusion
The magnitude of the Earth's magnetic field at Tucson, Arizona, is:
\[ B = 52 \, \mu T \]