Magnetic Fields

How Capacitors Store Electrical Energy

Problem. PS and PP are rates of energy dissipation in 3 resistors...

Problem

P_S and P_P are rates of energy dissipation in 3 resistors, when they are connected in series and in parallel respectively with some source of constant voltage. Let lowest resistance is known, r, 2^nd resistance is greater then the first on the factor of N. What is the resistance of the 3^rd resistor?
(a) Derive algebraic expression for the second resistance
and then
(b) use this expression to find numerical solution for the following data:
P_S=1 W, P_P=11 W, N=2, r=1Ω.

Solution:

V²/R_P=P_P

V²/R_S=P_S

R is 3^rd resistance.

R_S=r+Nr+R

1/R_P=1/r+1/(Nr) +1/R

R_P=1/( 1/r+1/(Nr) +1/R  )

P_P/P_S =R_S/R_P = R_S·(1/R_P)

P_P/P_S =R_S/R_P = (r+Nr+R)·( 1/r+1/(Nr) +1/R)

P_P/P_S=a

(r+Nr+R) (1/r+1/(Nr)+/R)=a

(1+N+R/r)(1+ 1/N + r/R) = a; Solve for x=R/r

R/r=x

P_P/P_S=a=11

11=(1+2+x)(1+1/2+1/x)

11=(3+x)(3/2+1/x)

11x=(3+x)(1.5x+1)

Solutions: x=3, x=2/3

R/r = 3

R=3·1Ω = 3Ω

Problem: PS and PP are rates of energy dissipation in two resistors, when they are connected in series and in parallel respectively with some source of constant voltage. Let lower resistance is known, r, what is resistance of another resistor?

Problem

P_S and P_P are rates of energy dissipation in two resistors, when they are connected in series and in parallel respectively with some source of constant voltage. Let lower resistance is known, r, what is resistance of another resistor?
(a) Derive algebraic expression for the second resistance
and then
(b) use this expression to find numerical solution for the following data:
P_S=12 W, P_P=54 W, r=1Ω.

Solution:

V²/R=P

R_S=R+r

1/R_P=1/R+1/r=(R+r) / (Rr)

R_P=Rr/(R+r)

V²/R_P=P_P

V²/R_S=P_S

R_S/R_P=P_P/P_S

(R+r) / (Rr/(R+r)) = (R+r)²/Rr=P_P/P_S=a

(R+r)²/Rr=a

(R+r)²=aRr

(R/r+1)²=aR/r

R/r = x

(x+1)²=ax

x²+2x+1=ax

x²+2x-ax=-1

x²+2(1-a/2)x=-1

x²+2(1-a/2)x+(1-a/2)²=-1+(1-a/2)²

(  x+(1-a/2)  )² = -1+(1-a/2)²

x+(1-a/2) = ± (  -1+(1-a/2)²  )^½

x=-(1-a/2) ± (  -1+(1-a/2)²  )^½

x=(a/2 - 1) ± ( (a/2-1)²-1  )^½

R/r =    (P_P/P_S)/2 – 1) ±  ((P_P/P_S)/2-1)²-1  )^½

R = r·(   (P_P/P_S)/2 - 1) ± ( ((P_P/P_S)/2-1)²-1  )^½ )

(a)        R = r·(   (P_P/P_S)/2 – 1) + ( ((P_P/P_S)/2-1)²-1  )^½ )

(P_P/P_S)/2 - 1) = (54 W/12 W)/2-1

(P_P/P_S)/2 - 1) = (  9 / 2) /2-1 = 9/4 – 4/4 = 5/4

R = 1Ω ·(5/4 + (25/16-1) ^½) = 1Ω ·(5/4 + (25/16-16/16) ^½) = 1Ω ·(5/4 + (9/16) ^½) = 1Ω ·(5/4 + ¾) = 1Ω ·(8/4) =2Ω

(b)  R = 2Ω

What is the resistance...

4. What is the resistance (R) of the resistor if the electric current (I) flowing through this resistor is equal to 2 A, and the electric potential (V_L) on the left terminal of this resistor is 4 V and the electric potential (V_r) on the right terminal of this resistor is 10 V. Which is direction of the electric current through this resistor, toward left or right?
V_r - V_L = ΔV_r,l = -I_r,l R
-I_r,l R = (V_r - V_L )

-I_r,l  = (V_r - V_L )/R

I_r,l  = (V_L – V_r )/R

 R = |(V_L – V_r )/ I_r,l |

R  = |(4V – 10V )/2A| = 3 Ω

The electric current flows from greater potential, 10V, right, to the lower potential, 4V, left: toward to left.

Exam 1. Problem 1

Exam 1. October 8, 2015, 8:00 AM
In solutions students should show:
a) given data by symbols,
b)   the unknown by symbol,
c) useful for this problem general formulae,
d) algebra job and final algebraic expression for the unknown,
e) substitution of symbols in the final algebraic expression by numbers with measurement units,
f) if it is required by a problem the final number with measurement units.

1. Electric field (E) accelerates a charged droplet from the zero speed to the final speed (V₁) of 8 m/s. What will be the final speed (V₂) if the electric field is decreased by a factor of 4, but all other parameters of this accelerator are not changed?

The electric field is decreased by a factor of 4 means: E₂= E/4, E₂/E=1/4

Potential energy of electric field, U=E∙d∙q
Kinetik Energy, K=½mv²

U=K

E∙d∙q = ½mv₁²,
where d is the traveling distance of the droplet in the electric field, q is the droplet’s electric charge.

E₂∙d∙q = ½mv₂²

(E₂∙d∙q)/ (E₁∙d∙q) =  (½mv₂²)/(½mv₁²)

E₂/ E =  v₂²/v₁²

v₂² = v₁² ∙E₂/ E

v₂ = v₁ ∙(E₂/ E)^1/2

v₂ = 8m/s ∙(1/4)^1/2 = 8m/s ∙(1/2) = 8m/s  /2 = 4m/s   

Practice Problems for the Exam 1.

 1. W_S is the rate of the energy dissipation in two resistors with resistances R₁ and R₂, when they are connected in series with a source of constant voltage. What is the rate of the energy dissipation in these two resistors when they and the source of constant voltage are connected in parallel?

2. W_S is the energy dissipation in 3 resistors, when they are connected in series with some source of constant voltage. Resistances are R₁, R₂, and R₃. What is the energy dissipation in these 3 resistors, when they are connected in parallel (W_P)?

3. R₁ and R₂ are resistances of two pieces of wire. The wires have different cross-sectional areas, and the length of the second wire is greater by the factor of N than the length of the first wire. The wires made from the same material. What is the ratio of m₁/m₂, where m₁ and m₂ are masses of these resistors? Derive algebraic expression for ratio m₁/m₂ in terms of R₁, R₂, and N.

4. A positive charge, Q₁, is located in the coordinates (x₁, y₁, z₁). Negative charge, -Q₂, is located in coordinates (x₂, y₂, z₂). Q₁ > Q₂. What are coordinates of a point on the straight line passing through the first and the second charge, where the electric field, created by these two charges, is equal to zero? Find algebraic expression in the terms of x₁, y₁, z₁, x₂, y₂, z₂, Q₁, and Q₂.

5. Two equivalent electric charges repel each other with the force F_I. What will be the interaction force F_F, if P% of one charge is transferred to the other?

6. U_S is total energy of two capacitors with capacitance C₁ and C₂, when they are connected in series with a source of constant voltage. What is total energy of these two capacitors, when they and the source of constant voltage are connected in parallel?

7. What is ratio of R_S/R_P for the N identical resistors, where R_S and R_P are their equivalent resistances when they are connected in series and in parallel respectively?

8. A positive charge Q₁ is located on the distance d₁ from some point in space. What is the distance (d₂) between the same point in space and the negative charge -Q₂, if the electric potentials of electric field, created by these two charges in this point and on the very far distance from these two charges are equal each other? 

9. An electric current I flows through a resistor with the resistance R from the resistor's terminal A to the terminal B. The electric potential in the point A is equal to V_A. What is the electric potential in the point B?

10. A capacitor with the capacitance C and the initial potential V totally discharges through some resistor and produces average current I in this resistor. What is the time of this discharge?

11. Five electric charges 10C, 11C, 12C, 13C, 14C are located on the X axis in coordinates 1m, 2m, 3m, 4m, and 5m respectively. There is a sphere with radius 1.1 m and which the center located in one of these electric charges. If the flux of the electric field, created by these electric charges, flowing through the surface of this sphere, equal to 39C/ε₀ ,  what is the coordinate of the sphere center? 

ε₀ = 8.9 × 10⁻¹² F/m

12. Two identical positive electric charges Q repel each other with the force F_I. When additional electric charge is added in the middle point between these two electric charges, magnitudes of net forces acting to old charges are F_F. Derive expression for additional electric charge in the terms of Q, F_I, F_F.

 13. Electric field E in an accelerator accelerates a charged droplet from the zero speed to the final speed V₁. To increase the final speed to V₂ electric field have to be increased by the factor of N. Derive expression for the factor N in terms of V₁ and V₂.

14. If the charge Q₁ is located on the vertex of acute angle of right triangle oppositely to the cathetus with the length d₁, and the charge Q₂ is located on the vertex of the acute angle, opposite to cathetus with the length d₂, then the net force F acts to the charge Q₃ located on the vertex of the right angle. Find algebraic expression for the magnitude of force F in terms of Q₁, Q₂, Q₃, d₁, and d₂.