Q. how to calculate net charge

Definition and goal. Net charge means the total electric charge of a system, obtained by algebraically adding all individual charges. The unit is the coulomb, abbreviated C.

Case 1, discrete charges. Steps to compute net charge when you have a finite set of point charges.

Step 1 Identify each charge value and its sign. Positive charges are positive numbers. Negative charges are negative numbers.

Step 2 Convert all values to the same unit, coulombs, if they are not already.

Step 3 Add the charges algebraically using the formula

\[ Q_{\mathrm{net}} = \sum_{i} q_{i} \]

Step 4 Interpret the sign of the result. A positive \( Q_{\mathrm{net}} \) means net positive charge. A negative \( Q_{\mathrm{net}} \) means net negative charge. Zero means the system is electrically neutral.

Example for discrete charges. Suppose three charges are given: \( q_{1}=+2.0\ \mathrm{C} \), \( q_{2}=-0.75\ \mathrm{C} \), and \( q_{3}=+0.25\ \mathrm{C} \).

Step 1 Verify units. All are in coulombs.

Step 2 Sum them algebraically.

\[ Q_{\mathrm{net}} = q_{1}+q_{2}+q_{3} = (+2.0)+(-0.75)+(+0.25) \]

\[ Q_{\mathrm{net}} = 2.0-0.75+0.25 \]

\[ Q_{\mathrm{net}} = 1.5\ \mathrm{C} \]

Step 3 Interpret. The net charge is \( +1.5\ \mathrm{C} \), so the system has a positive net charge of one point five coulombs.

Case 2, continuous charge distributions. If charge is distributed continuously, compute net charge by integrating the charge density over the region that contains charge.

Use the appropriate density and integral.

For a line distribution with linear charge density \( \lambda(x) \):

\[ Q = \int \lambda(x)\, dx \]

For a surface distribution with surface charge density \( \sigma(\mathbf{r}) \):

\[ Q = \int \sigma(\mathbf{r})\, dA \]

For a volume distribution with volume charge density \( \rho(\mathbf{r}) \):

\[ Q = \int \rho(\mathbf{r})\, dV \]

Example for a uniform rod. A rod of length \( L=0.50\ \mathrm{m} \) has constant linear charge density \( \lambda=2.0\ \mathrm{C/m} \). Compute the total charge.

\[ Q = \int_{0}^{L} \lambda\, dx = \lambda L = (2.0\ \mathrm{C/m})(0.50\ \mathrm{m}) \]

\[ Q = 1.0\ \mathrm{C} \]

Units and sign. Always keep units of coulombs and carry the sign through the arithmetic. If the density function can be negative at some points, the integral already accounts for that by algebraic addition.

Summary checklist for any problem asking for net charge.

1. List all charges or the charge density and domain. 2. Convert to consistent units, coulombs. 3. For discrete charges, compute \( Q_{\mathrm{net}}=\sum_i q_i \). 4. For continuous distributions, compute the appropriate integral. 5. Report the numerical value with units and the sign, and state whether the net charge is positive, negative, or zero.