Why can positively and negatively charged rods attract an object with no charge?
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Something that maintains a positive charge has more positively charged particles, and something that maintains a negative charge has more particles that are negatively charged. In that way, something that has no charge, or is neutral, keeps the two in balance. An electrostatic charge is created by these particles, and how many of each type are involved will determine if that field is negative, positive, or neutral.
Opposite fields attract and like fields repel. That's because something that is negatively charged is trying to "take" from the positive to balance itself out, and vice versa. Two negatively charged things don't want each-other because there are no positives to mooch, and two positives push against each-other for the same reason.
Why, then, would something neutral be attracted? In your question, the negatively or positively charged rods come near something neutral. This creates a charge in the neutral object that is equal and opposite to the one it has encountered, and as we know, opposites attract. This charging process is called induction.
Think of it this way: the positive rods are trying to give away protons, and the negative rods are trying to get rid of electrons. That's why they like each-other. Each has what the other wants. In a way, a neutral object is the best of both worlds. To something that is negatively charged, it has fewer electrons, and to something positively charged, it has fewer protons. It is balanced and can be drawn either way because the other two are out of balance.
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