The magnitude of electric field due to a point charge 2q, at distance r is E. Then the magnitude of electric field due to a uniformly charged thin spherical shell of radius R with total charge q at a distance – (r >> R) will be

Step 1: Electric Field due to a Point Charge

The electric field, E, due to a point charge Q at a distance r is given by Coulomb's Law:

E = k * Q / r²    

Where:

• k is Coulomb's constant (9 × 10⁹ N·m²/C²).
• Q is the magnitude of the charge.
• r is the distance from the charge.

Step 2: Electric Field due to the Spherical Shell

For a uniformly charged thin spherical shell with a total charge q and radius R, the electric field outside the shell (at a distance r from the center, where r > R) behaves as if all the charge were concentrated at the center of the shell. Thus, the electric field at a distance r from the center of the shell is given by the formula:

E_shell = k * q / r²    

Where:

• q is the total charge on the spherical shell.
• r is the distance from the center of the spherical shell.

Step 3: Calculate the Electric Field at Distance r

Given that r ≫ R (the distance is much greater than the radius of the shell), we can directly use the formula for the electric field due to the spherical shell:

E_shell = k * q / r²    

This electric field is similar to that of a point charge q located at the center of the spherical shell.

Step 4: Relate the Electric Field to the Point Charge

Now, we are given that the electric field due to a point charge of magnitude 2q at a distance r is E:

E = k * (2q) / r²    

We can now express the electric field due to the spherical shell in terms of this known value of E.

Step 5: Finding the Electric Field due to the Shell

The electric field due to the spherical shell at distance r can be written as:

E_shell = k * q / r²    

We can express q in terms of 2q (since the field due to the point charge of 2q is E):

E_shell = (k * q / r²) = (1/2) * (k * (2q) / r²)    

Substituting E for (k * (2q) / r²), we get:

E_shell = (1/2) * E    

Conclusion

Thus, the magnitude of the electric field due to a uniformly charged thin spherical shell at a distance r (where r ≫ R) is:

        E_shell = 1/2 * E