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The surface charge density of a thin charged disc of radius R is $$\sigma$$. The value of the electric field at the centre of the disc is $$\frac{\sigma}{2\epsilon_0}$$. With respect to the field at the centre, the electric field along the axis at a distance R from the centre of the disc:
The electric field at a point along the axis of a uniformly charged disc is given by the formula:
$$E(x) = \frac{\sigma}{2\epsilon_0} \left(1 - \frac{x}{\sqrt{x^2 + R^2}}\right)$$
At the center of the disc, where $$x = 0$$, the electric field is:
$$E(0) = \frac{\sigma}{2\epsilon_0} \left(1 - \frac{0}{\sqrt{0^2 + R^2}}\right) = \frac{\sigma}{2\epsilon_0} (1 - 0) = \frac{\sigma}{2\epsilon_0}$$
This matches the given value. Now, we need to find the electric field at a distance $$R$$ from the center along the axis, so we substitute $$x = R$$:
$$E(R) = \frac{\sigma}{2\epsilon_0} \left(1 - \frac{R}{\sqrt{R^2 + R^2}}\right)$$
Simplify the denominator:
$$\sqrt{R^2 + R^2} = \sqrt{2R^2} = R\sqrt{2}$$
So,
$$E(R) = \frac{\sigma}{2\epsilon_0} \left(1 - \frac{R}{R\sqrt{2}}\right) = \frac{\sigma}{2\epsilon_0} \left(1 - \frac{1}{\sqrt{2}}\right)$$
Let $$E_c = E(0) = \frac{\sigma}{2\epsilon_0}$$ denote the field at the center. The reduction in the electric field at $$x = R$$ compared to the center is $$E_c - E(R)$$. The percentage reduction with respect to $$E_c$$ is:
$$\text{Percentage reduction} = \frac{E_c - E(R)}{E_c} \times 100\%$$
Substitute the expressions:
$$\frac{E_c - E(R)}{E_c} = \frac{\frac{\sigma}{2\epsilon_0} - \frac{\sigma}{2\epsilon_0} \left(1 - \frac{1}{\sqrt{2}}\right)}{\frac{\sigma}{2\epsilon_0}}$$
The common factor $$\frac{\sigma}{2\epsilon_0}$$ cancels out:
$$= \frac{1 - \left(1 - \frac{1}{\sqrt{2}}\right)}{1} = \frac{1 - 1 + \frac{1}{\sqrt{2}}}{1} = \frac{1}{\sqrt{2}}$$
Therefore,
$$\text{Percentage reduction} = \frac{1}{\sqrt{2}} \times 100\%$$
Calculate the numerical value:
$$\frac{1}{\sqrt{2}} \approx 0.7071$$
$$0.7071 \times 100\% = 70.71\%$$
Rounding to one decimal place, this is approximately 70.7%.
Hence, the electric field reduces by 70.7% with respect to the field at the center.
So, the answer is Option A.
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