Question

two-magnet system (north poles facing each other)
before
after
charged balloon-hair system
how did the potential energy of each system change?

• the magnetic potential energy of the two-magnet system
• the electric potential energy of the balloon-hair system

options: increased, decreased, stayed the same

Explanation:

For the two - magnet system:

Step1: Analyze magnet interaction

Like magnetic poles (north - north here) repel each other. In the "Before" state, the magnets are separated, and in the "After" state, they are closer (since they are pushed towards each other but maybe the distance decreases? Wait, no, when like poles face each other, to bring them closer, we have to do work against the repulsive force? Wait, no, actually, if the magnets move towards each other (even though they repel), wait, looking at the images: in the "Before" the two magnets are held with some distance, in the "After" they are closer. But when like poles are facing, the potential energy of a magnetic system with like poles closer is higher? Wait, no. The formula for magnetic potential energy in a simple case: when two magnets with like poles are brought closer, the potential energy increases because we are doing work against the repulsive force. Wait, or is it the other way? Let's think about conservative forces. The magnetic force between like poles is repulsive. So if the distance between them decreases (they are brought closer), the work done by the external force (to bring them closer against repulsion) is positive, so the potential energy (which is related to the work done by the conservative force, $U = -W_{conservative}$) will increase. Wait, no: the change in potential energy $\Delta U=W_{external}$ when the system is isolated. So if we push the two like - pole magnets closer, we do positive work, so $\Delta U$ is positive, meaning potential energy increases? Wait, no, maybe I got it wrong. Let's take a simple case: two positive charges (analogous to like magnetic poles) are separated by distance $r_1$, then brought to $r_2

Step2: Conclusion for magnet system

Since the two magnets have north poles facing (like poles), and if the distance between them decreases (from "Before" to "After"), the magnetic potential energy increases. Wait, but maybe the "After" shows them moving apart? No, the image shows them closer. So the magnetic potential energy of the two - magnet system increased? Wait, no, maybe I made a mistake. Wait, let's think about the balloon - hair system first.

For the balloon - hair syst…

Step1: Analyze electric interaction

A charged balloon and hair (after rubbing, the balloon and hair have opposite charges, since charging by friction: the balloon gains electrons, hair loses electrons, so balloon is negative, hair is positive). Opposite charges attract. In the "Before" state, maybe the balloon is near the hair, and in the "After" state, the hair is more attracted (the hair strands are moving towards the balloon? Wait, the "After" image shows the hair more "sticking up" or attracted? Wait, the problem's "After" for the balloon - hair system: the hair looks more attracted. When opposite charges are brought closer (or the distance between the charged balloon and the hair decreases), the electric potential energy of a system of two opposite charges is $U=-\frac{kq_1q_2}{r}$ (since $q_1$ and $q_2$ have opposite signs). So when $r$ decreases, $U$ becomes less negative (or increases? Wait, no: if $q_1$ is negative and $q_2$ is positive, $U = \frac{k(-|q_1|)(|q_2|)}{r}=-\frac{k|q_1q_2|}{r}$. When $r$ decreases, the magnitude of $U$ increases, but since it's negative, it becomes less negative (i.e., increases). Wait, for example, if $r = 2$, $U=-5$; if $r = 1$, $U=-10$? No, wait, no: $U=-\frac{k|q_1q_2|}{r}$, so as $r$ decreases, $|U|$ increases, so $U$ becomes more negative? Wait, I messed up the formula. The correct formula for electric potential energy between two point charges is $U=\frac{kq_1q_2}{r}$. So if $q_1$ and $q_2$ are opposite (one positive, one negative), $q_1q_2$ is negative, so $U$ is negative. When the distance $r$ between them decreases, $U=\frac{kq_1q_2}{r}$: since $q_1q_2<0$, and $r$ decreases, $U$ becomes more negative (i.e., decreases). Wait, that makes sense. Because opposite charges attract, so when they move closer (due to the attractive force), the force does positive work, so the potential energy decreases (since $\Delta U=-W_{conservative}$, and $W_{conservative}$ is positive when the force moves the objects closer). So for the balloon - hair system (opposite charges), when they move closer (the hair is attracted to the balloon, so distance between charged parts decreases), the electric potential energy decreases? Wait, no, let's take an example. Suppose $q_1 = +1$, $q_2=-1$, $k = 1$. At $r = 2$, $U=\frac{(1)(-1)}{2}=-\frac{1}{2}$. At $r = 1$, $U=\frac{(1)(-1)}{1}=-1$. So as $r$ decreases from 2 to 1, $U$ decreases from - 0.5 to - 1. So yes, for opposite charges, decreasing the distance decreases the electric potential energy (makes it more negative).

Now, going back to the magnet system: like poles (north - north), so $q_1$ and $q_2$ are like (both north, analogous to both positive charges). The magnetic potential energy (analogous to electric potential energy for like charges) is $U=\frac{k'm_1m_2}{r}$ (where $m$ is magnetic pole strength, $k'$ is a constant). So when $r$ decreases (magnets are brought closer), $U$ increases (since $m_1m_2>0$ for like poles). So in the two - magnet system, if the distance between the north poles decreases (from "Before" to "After"), the magnetic potential energy increases.

Wait, but the "After" image for the magnets: are they closer or farther? The "Before" has two magnets held with some space, "After" shows them closer (the hands move them closer). So for the two - magnet system (like poles), bringing them closer increases the magnetic potential energy.

For the balloon - hair system: opposite charges, so when they are attracted (distance decreases), the electric potential energy decreases.

Answer:

• The magnetic potential energy of the two - magnet system: increased
• The electric potential energy of the balloon - hair system: decreased