A coil of wire with cross-sectional area 0.50 `m m^(2)` weighs 75 N in air and 65 N in water. The length of the coil in cm is (Take g = 10 m/`s^(2)`)

Options

A. `(10^(2))/(50)`
B. `(10^(20))/(0.50)`
C. `(10^(6))/(5)`
D. `(10^(5))/(0.0050)`

Explanation

To find the length of the coil, we can use the information provided about the weight of the coil in air and water. Here’s a step-by-step solution:

### Step 1: Understand the problem
We know the weight of the coil in air (W_air = 75 N) and in water (W_water = 65 N). The difference in weight when submerged in water is due to the buoyant force acting on it.

### Step 2: Calculate the buoyant force
The buoyant force (B) can be calculated as:
\[ B = W_{air} - W_{water} \]
\[ B = 75 \, \text{N} - 65 \, \text{N} = 10 \, \text{N} \]

### Step 3: Use Archimedes' principle
According to Archimedes' principle, the buoyant force is equal to the weight of the fluid displaced by the coil. The weight of the water displaced can be expressed as:
\[ B = \text{Volume}_{displaced} \times \text{Density}_{water} \times g \]
Where:
- \(\text{Density}_{water} = 1000 \, \text{kg/m}^3\)
- \(g = 10 \, \text{m/s}^2\)

### Step 4: Relate volume to the cross-sectional area and length
The volume of the coil can be expressed in terms of its cross-sectional area (A) and length (L):
\[ \text{Volume} = A \times L \]
Given that the cross-sectional area \(A = 0.50 \, \text{mm}^2 = 0.50 \times 10^{-6} \, \text{m}^2\).

### Step 5: Set up the equation
Now we can substitute the volume into the buoyant force equation:
\[ B = A \times L \times \text{Density}_{water} \times g \]
Substituting the known values:
\[ 10 \, \text{N} = (0.50 \times 10^{-6} \, \text{m}^2) \times L \times (1000 \, \text{kg/m}^3) \times (10 \, \text{m/s}^2) \]

### Step 6: Solve for L
Now we can solve for L:
\[ 10 = (0.50 \times 10^{-6}) \times L \times 10000 \]
\[ 10 = 5 \times 10^{-3} \times L \]
\[ L = \frac{10}{5 \times 10^{-3}} \]
\[ L = 2000 \, \text{m} \]

### Step 7: Convert L to cm
To convert L from meters to centimeters:
\[ L = 2000 \, \text{m} = 2000 \times 100 \, \text{cm} = 200000 \, \text{cm} \]

### Final Answer
The length of the coil is \(200000 \, \text{cm}\).

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