= Solution
For the <polybutadiene> chain in <1,4-dioxane>, define $R_x$ from the negative end charge toward the positive one. The <electric dipole> energy is $-eER_x$, so the force conjugate to end separation is $eE$, even though the two endpoint <forces> are opposite. Balancing it against the <entropic spring> force gives
$$
\langle R_x\rangle=\frac{Na^2eE}{3k_BT},\qquad T=299.65\,\mathrm K.
$$
Numerically,
$$
\boxed{\langle R_x\rangle=46.5\,\mathrm{nm},\qquad R_{\rm rms,0}=a\sqrt N=60.0\,\mathrm{nm},\qquad\frac{\langle R_x\rangle}{R_{\rm rms,0}}=0.775.}
$$
The zero-field rms of just the $x$ component is $60/\sqrt3=34.6\,\mathrm{nm}$, a different comparison. The mean extension is still much smaller than the contour length $Na=6\,\mathrm{\mu m}$, consistent with the Gaussian <ideal chain> force law. This is an <end-charged ideal polymer in an electric field>; the direct endpoint <Coulomb interaction> is omitted as stipulated.
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