Q.Figures 2.8(a) and

Since $V\propto 1/r$ for a point charge, the point nearer a positive charge is at higher potential, and the point nearer a negative charge is at lower (more negative) potential. Using $U=qV$ and $W_{\text{field}}=-\Delta U$: (a) $V_P-V_Q>0$ and $V_B-V_A>0$;

(b) the potential-energy difference of a small negative charge is positive both between Q and P and between A and B;

(c) the field does negative work moving a positive charge from Q to P; …

Concept understanding

Electric potential is a scalar equal to the work per unit positive charge to bring it from infinity. For a point charge,

so for a positive charge $V$ is positive and decreases with $r$ (P nearer $\Rightarrow V_P>V_Q$), while for a negative charge $V$ is negative and becomes less negative with $r$ (B farther $\Rightarrow V_B>V_A$). The potential energy of a test charge is $U=qV$, and the work done by the field is $W_{\text{field}}=-\Delta U$.

Part-by-part

(a) For the positive charge P is closer than Q, so $V_P>V_Q$ and $V_P-V_Q>0$. For the negative charge A is closer than B, so $V_A$ is more negative, giving $V_B>V_A$ and $V_B-V_A>0$.

(b) For a small negative test charge, $U=qV$ with $q<0$, so $U$ has the sign opposite to $V$.

• Q$\to$P: $V_P>V_Q\Rightarrow U_P<U_Q$, so $U_Q-U_P>0$ — positive.
• A$\to$B: $V_B>V_A\Rightarrow U_B<U_A$, so $U_A-U_B>0$ — positive.

(c) Moving a positive charge from Q (lower $V$) to P (higher $V$) is against the field, so the field does negative work: $W_{\text{field}}=U_Q-U_P<0$. …

Method: Determining Signs of Potential, Potential Energy, and Work from a Field-Line Picture

This method applies to conceptual questions that show field lines of one or more point charges and ask only for the sign of a potential difference, potential-energy difference, or work done — not a numeric value.

Steps

Step 1: Rank the points by potential using $V \propto 1/r$

For a positive source charge, potential is positive and falls off with distance — the point nearer the charge is at the higher potential. For a negative source charge, potential is negative and rises (becomes less negative) with distance — the point farther from the charge is at the higher potential. Use this to decide the sign of any potential difference asked for.

Step 2: Convert to potential-energy sign using $U = qV$

For a positive test charge, $U$ has the same sign as $V$. For a negative test charge, $U$ has the opposite sign to $V$ — this sign flip is the most common place to make an error, so state it explicitly every time.

Step 3: Get the sign of work done by the field using $W_{\text{field}} = -\Delta U$ …