Lecture 20: FRI 09 OCT Circuits II Physics 2113 Jonathan Dowling

How to Solve Multi-Loop Circuits

Monster Maze If all resistors have a resistance of 4, and all batteries are ideal and have an emf of 4V, what is the current through R? Step 1: Find loop that only goes through R and through batteries but no other resistors. Step 2: Assume i is clockwise on the loop and walk the loop using the loop rules. Step 3: Solve for i. Step 4: If i is negative, reverse the direction of the current in the diagram.

What If We Chose Wrong Direction for i ??? If all resistors have a resistance of 4, and all batteries are ideal and have an emf of 4V, what is the current through R? Step 1: Find loop that only goes through R and through batteries but no other resistors. Step 2: Assume i is counter- clockwise on the loop and walk the loop using the loop rules. Step 3: Solve for i. Step 4: If i is negative, reverse the direction of the current in the diagram.

One Battery? Simplify! Resistors Capacitors Key formula: V=iR Q=CV In series: same current dQ/dt same charge Q R eq =∑R j 1/C eq = ∑1/C j In parallel: same voltage same voltage 1/R eq = ∑1/R j C eq =∑C j P = iV = i 2 R = V 2 /R U = QV/2 = Q 2 /2C = CV 2

Too Many Batteries? Loop & Junction! One Battery: Simplify First Three Batteries: Straight to Loop & Junction Rules: Circuit CAN’T Be Simplified Because R 1 and R 2 are neither in Series Nor in Parallel Because Batteries Between Them.

Around every loop moving in direction of current add + E if you cross a battery from minus to plus, – E if plus to minus, and –iR for each resistor. Then sum to Zero: + E 1 – E 2 – iR 1 – iR 2 = 0. Assume current and loop clockwise. Apply Loop Rule R1R1 R2R2 E1E1 E2E2 +–+– +–+– Conservation of ENERGY!

Apply Junction Rule At every junction sum the ingoing currents and outgoing currents and set them equal. i 1 = i 2 + i 3 i1i1 i2i2 i3i3 Conservation of CHARGE!

Equations to Unknowns Continue applying loop and junction until you have as many equations as unknowns! + E 1 – E 2 – i 1 R 1 – i 2 R 2 = 0 and i 1 = i 2 + i 3 Solve for i 2, i 3 Given: E 1, E 2, i 1, R 1, R 2

Example Find the equivalent resistance between points (a) F and H and (b) F and G. (Hint: For each pair of points, imagine that a battery is connected across the pair.) Compile R ’ s in Series Compile equivalent R ’ s in Parallel F H F H F H Slide Rule Series Parallel

Example Assume the batteries are ideal, and have emf E 1 =8V, E 2 =5V, E 3 =4V, and R 1 =140  R 2 =75  and  R 3 =2. What is the current in each branch? What is the power delivered by each battery? Which point is at a higher potential, a or b? Apply loop rule three times and junction rule twice.

Example What ’ s the current through resistor R 1 ? What ’ s the current through resistor R 2 ? What ’ s the current through each battery? Apply loop rule three times and junction rule twice.

Too many batteries: loop and junction! There are three loops and two junctions. Assume all currents are clockwise. Both junctions give same result: Walk three loops clockwise from a to a: Three equations and three unknowns i 1, i 2, i 3. Tough in general but trick here is to realize red and green loop equations depend on only one of the i’s. Solve for i 1 and i 3 and use junction to get i 2. Finally reverse i 3.

E 1 =8V, E 2 =5V, E 3 =4V, and R 1 =140  R 2 =75  and  R 3 =2. What is the current in each branch? What is the power delivered by each battery? Which point is at a higher potential, a or b? What ’ s the current through resistor R 1 What ’ s the current through resistor R 2 ? What ’ s the current through each battery?

Non-Ideal Batteries You have two ideal identical batteries, and a resistor. Do you connect the batteries in series or in parallel to get maximum current through R? Does the answer change if you have non-ideal (but still identical) batteries? Apply loop and junction rules until you have current in R.

More Light Bulbs If all batteries are ideal, and all batteries and light bulbs are identical, in which arrangements will the light bulbs as bright as the one in circuit X? Does the answer change if batteries are not ideal? Calculate i and V across each bulb. P = iV = “ brightness ” or Calculate each i with R ’ s the same: P = i 2 R