In a capillary tube of radius 'R' a straight thin metal wire of radius 'r' (`Rgtr)` is inserted symmetrically and one of the combination is dipped vertically in water such that the lower end of the combination Is at same level . The rise of water in the capillary tube is [T=surface tensiono of water `rho` =density of water ,g =gravitational acceleration ]

Options

A. `T/((R + r)rhog)`
B. `(2T)/((R + r) rho g)`
C. `(2 T)/((R - r)rhog)`
D. `((R - r)rhog)/(T)`

Answer

To solve the problem of the rise of water in a capillary tube with a thin wire inserted symmetrically, we can follow these steps:

### Step 1: Understand the Setup
We have a capillary tube of radius \( R \) and a thin metal wire of radius \( r \) inserted symmetrically into the tube. The system is dipped in water, and we need to find the height \( h \) to which the water rises in the capillary tube.

**Hint:** Visualize the setup and identify the forces acting on the water due to surface tension and pressure.

### Step 2: Identify the Forces
The surface tension \( T \) acts at two surfaces: at the interface of the wire and the water, and at the interface of the water and the capillary tube. The total upward force due to surface tension can be expressed as:
\[ F_{\text{surface tension}} = 2\pi r T + 2\pi R T \]
where \( 2\pi r T \) is the force due to the wire and \( 2\pi R T \) is the force due to the capillary tube.

**Hint:** Remember that the surface tension acts along the circumference of the surfaces in contact with the liquid.

### Step 3: Calculate the Weight of the Water Column
The weight of the water column that rises to height \( h \) in the capillary tube can be expressed as:
\[ F_{\text{weight}} = \rho V g = \rho (\pi (R^2 - r^2) h) g \]
where \( V \) is the volume of the water column, \( \rho \) is the density of water, and \( g \) is the acceleration due to gravity.

**Hint:** Use the formula for the volume of a cylinder to find the weight of the water.

### Step 4: Set Up the Equilibrium Condition
At equilibrium, the upward force due to surface tension equals the downward force due to the weight of the water:
\[ 2\pi r T + 2\pi R T = \rho (\pi (R^2 - r^2) h) g \]

**Hint:** This is a balance of forces; one side represents the upward force and the other side represents the downward force.

### Step 5: Simplify the Equation
We can simplify the equation by canceling \( \pi \) from both sides:
\[ 2rT + 2RT = \rho (R^2 - r^2) h g \]
Now, we can isolate \( h \):
\[ h = \frac{2T (r + R)}{\rho (R^2 - r^2) g} \]

**Hint:** Rearranging the equation helps in isolating the variable we want to solve for.

### Step 6: Final Expression
Thus, the height \( h \) to which the water rises in the capillary tube is given by:
\[ h = \frac{2T (R + r)}{\rho (R^2 - r^2) g} \]

**Hint:** Ensure that the final expression is dimensionally consistent and matches the physical scenario.

### Summary
The rise of water in the capillary tube with a wire inserted is determined by the balance of surface tension forces and the weight of the water column. The derived formula gives a clear relationship between the physical parameters involved.