# What does negative electrical energy signify?

When we derive the formula for potential energy caused by the torque of a dipole in uniform electrical field we get $$U = -pE \cos \theta$$. And my textbook tells me that the when the dipole is kept parallel to the electric field, the angle made is zero $$\cos\theta = 1$$, thus the potential energy $$U$$ is $$-pE$$. The textbook also tells me that this is the minimum energy attained by a dipole in external electric field ($$U = -pE$$) and the configuration is stable. But since energy is a scalar quantity, it isn't supposed to have direction, So, what does the negative symbol signify?

Absolute values of the potential energies of systems do not have any physical meaning. It is the change in potential energy that has a physical meaning.

When the potential energy of a dipole system is derived, the following approach is used: $$U_f - U_i = \int dW = \int_{\theta_i}^{\theta_f} \tau\cdot d\theta$$ $$U_f - U_i = pE \int_{\theta_i}^{\theta_f}\sin\theta\cdot d\theta,$$ and consequently, $$\boxed{\Delta U= -pE(\cos\theta_f - \cos\theta_i).}$$

From here, a reference configuration is chosen for the dipole system such that $$U_i = 0 \ \text{at} \ \theta_i = 90^{\circ}$$, purely for convenience, which gives the formula $$U = -pE\cdot\cos\theta$$.

Naturally, any positive or negative signs arise only due to this choice of convention and do not possess any meaning as such.

Hope this helps.

• So negative PE signifies the decrease in potential energy from the chosen reference configuration right? Commented May 29, 2022 at 9:09
• Yes. For example, $U = mgy$ is a formula obtained only when you define $U = 0$ at $y = 0$ (usually the ground). If you go below $y= 0$, $U$ is negative only relative to the chosen configuration of $y = 0$. Commented May 29, 2022 at 9:14
• Wow, thank you so much :) Commented May 29, 2022 at 9:33