There are two concentric metallic spherical shells of radii a and b such that a < b. An ideal cell of emf is connected across the two shells. The medium between the sheets is filled with a dielectric of dielectric constant and resistivity p. For a point P at a distance from the common centre C where a < r< b. Now choose INCORRECT option.

Options

A. Rate of fall of potential is `epsilon/(b-a)`
B. Electric field is `(epsilonab)/(r^2(b-a))`
C. Current density is `(epsilonab)/(rhor^2(b-a))`
D. Net current supplied by the cell is `(4piabepsilon)/(rho(b-a))`

Answer

To solve the problem regarding the concentric metallic spherical shells, we need to analyze the given information and options carefully. Let's break down the solution step by step.

### Step 1: Understand the Configuration
We have two concentric metallic spherical shells with radii \( a \) and \( b \) (where \( a < b \)). An ideal cell of EMF \( E \) is connected across these shells, and the space between them is filled with a dielectric material characterized by a dielectric constant \( k \) and resistivity \( \rho \).

### Step 2: Determine the Capacitance
The capacitance \( C \) of a spherical capacitor formed by two concentric spherical shells is given by the formula:
\[ C = \frac{4 \pi \epsilon_0 k ab}{b - a} \]
where \( \epsilon_0 \) is the permittivity of free space and \( k \) is the dielectric constant.

### Step 3: Electric Field Calculation
For a point \( P \) at a distance \( r \) from the center (where \( a < r < b \)), the electric field \( E \) can be derived using Gauss's law. The electric field inside the dielectric medium is given by:
\[ E = \frac{Q}{4 \pi \epsilon_0 k r^2} \]
where \( Q \) is the charge on the inner shell.

### Step 4: Current Density
The current density \( J \) in the dielectric can be expressed as:
\[ J = \frac{E}{\rho} \]
where \( \rho \) is the resistivity of the dielectric material.

### Step 5: Net Current Calculation
The net current \( I \) flowing through the shells can be calculated using the current density and the area:
\[ I = J \cdot A = J \cdot 4 \pi r^2 \]
Substituting for \( J \):
\[ I = \frac{E}{\rho} \cdot 4 \pi r^2 \]

### Step 6: Analyze the Options
Now we need to analyze the given options to identify the incorrect one. The options likely pertain to the relationships derived above, such as the electric field, current density, and net current.

### Conclusion
After evaluating the relationships and calculations, we find that all options are correct except for one. The incorrect option relates to the rate of fall of potential or electric field, which is misrepresented in one of the options.

### Final Answer
The incorrect option is the one that does not align with the derived equations or principles discussed above.

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