An electron with mass \(m_e\) and a proton with mass \(m_p\) are each accelerated from rest through electric potential differences \(V_e\) and \(V_p\), respectively. Each particle carries a charge of magnitude \(e\). If the ratio of the electron's final de Broglie wavelength to the proton's final de Broglie wavelength is \(\dfrac{\lambda_e}{\lambda_p} = 2\), what is the ratio \(\dfrac{V_e}{V_p}\) of the accelerating potential differences?