A current canying straight wire is placed along east-west and current is passed through it eastward . The direction of the force act on it due to horizontal component of earth's magnetic field is

Options

A. due west
B. due south
C. vertically upwards
D. vertically downwards

Answer

To solve the problem, we need to determine the direction of the force acting on a current-carrying wire due to the horizontal component of the Earth's magnetic field. Let's break it down step by step:

### Step 1: Identify the Directions
- The wire is placed along the east-west direction, with the current flowing eastward.
- Therefore, we can denote:
- East direction as positive x-axis.
- West direction as negative x-axis.
- North direction as positive y-axis.
- South direction as negative y-axis.

### Step 2: Determine the Magnetic Field Direction
- The horizontal component of the Earth's magnetic field generally points towards the geographic north.
- Thus, we can denote the magnetic field \( \mathbf{B} \) as pointing in the positive y-direction (north).

### Step 3: Use the Right-Hand Rule
- The force \( \mathbf{F} \) on a current-carrying conductor in a magnetic field is given by the equation:
\[ \mathbf{F} = I (\mathbf{L} \times \mathbf{B}) \]
where:
- \( I \) is the current,
- \( \mathbf{L} \) is the length vector of the wire (in the direction of the current),
- \( \mathbf{B} \) is the magnetic field vector.

### Step 4: Define the Vectors
- The current \( I \) flows eastward, so we can represent the length vector \( \mathbf{L} \) as:
\[ \mathbf{L} = L \hat{i} \]
where \( \hat{i} \) is the unit vector in the x-direction (east).
- The magnetic field \( \mathbf{B} \) points north, so we can represent it as:
\[ \mathbf{B} = B \hat{j} \]
where \( \hat{j} \) is the unit vector in the y-direction (north).

### Step 5: Calculate the Cross Product
- Now we calculate the cross product \( \mathbf{L} \times \mathbf{B} \):
\[ \mathbf{F} = I (\mathbf{L} \times \mathbf{B}) = I (L \hat{i} \times B \hat{j}) = I LB (\hat{i} \times \hat{j}) \]
- The cross product \( \hat{i} \times \hat{j} \) gives \( \hat{k} \), which points in the positive z-direction (upward).

### Step 6: Conclusion
- Therefore, the direction of the force \( \mathbf{F} \) acting on the wire due to the horizontal component of the Earth's magnetic field is vertically upward.

### Final Answer
The direction of the force acting on the wire is **vertically upward**.

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