A parallel-plate capacitor with plate dimensions of length \(L\) and width \(w\), separated by distance \(d\), is connected across an ideal battery that maintains a constant potential difference \(V_0\). An uncharged dielectric slab of dielectric constant \(\kappa\), length \(L\), width \(w\), and thickness \(d\) is initially outside the capacitor. An external agent slowly inserts the slab at constant velocity into the space between the plates until it completely fills the capacitor. In terms of the given quantities and fundamental constants, what is the total work \(W_{\text{ext}}\rangle done by the external agent on the dielectric slab during this process?